Subgroup temperaments: Difference between revisions

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A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p-limit'' group.  
{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.  


Below are some subgroups and temperaments for them. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].
For temperaments that omit various prime harmonics, see:
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).


== No-elevens subgroup ==
Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].


=== [[Ultrapyth]] ===
This page uses an extended [[wart notation]] where squares and cubes of a prime basis element are denoted by ′ and ″, respectively, to reduce the use of temporary warts.  
: <small>''For full 13-limit extensions, see [[Archytas clan #Ultrapyth]].''</small>


Subgroup: 2.3.5.7.13
== 2.3.… subgroups ==
=== Shrub ===
Named by [[User: VectorGraphics|Vector]] in 2026, shrub is a [[restriction]] of [[diaschismic family #Diaschismic|diaschismic]] which omits the tritone to produce a [[5L 2s|diatonic]] scale. True to its name, it generates a [[neogothic major and minor|shrubmajor]]{{idio}} third (~425{{c}}) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.  


[[Comma list]]: 64/63, 91/90, 4394/4375
It can also admit a [[41/1|41st harmonic]] by tempering out [[82/81]] (same mapping as [[reversed meantone]]). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error.


[[Gencom]]: [2 4/3; 64/63 91/90 4394/4375]
==== 2.3.25 subgroup ====
[[Subgroup]]: 2.3.25


[[Gencom|Gencom mapping]]: [{{val|1 2 8 2 0 11}}, {{val|0 -1 -14 2 0 -18}}]
[[Comma list]]: 2048/2025


[[Mapping|Sval mapping]]: [{{val|1 2 8 2 11}}, {{val|0 -1 -14 2 -18}}]
{{Mapping|legend=3| 1 0 11 | 0 1 -4 }}
: mapping generators: ~2, ~3


[[Tp tuning|POL2 generator]]: ~4/3 = 486.255
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8214{{c}}, ~3/2 = 704.2059{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9585{{c}}


{{Val list|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=1| 12, 17, 46, 63, 80, 143c′ }}


[[Tp tuning #T2 tuning|RMS error]]: 2.318 cents
[[Badness]] (Sintel): 0.234


=== Sensi (aka Sensation) ===
===== 2.3.25.23 subgroup =====
: <small>''For full 13-limit extensions, see [[Sensipent family]] or [[Sensi extensions]].''</small>
Subgroup: 2.3.25.23


Subgroup: 2.3.5.7.13
Comma list: 576/575, 736/729


[[Comma list]]: 91/90, 126/125, 169/168
{{Mapping|legend=2| 1 0 11 -5 | 0 1 -4 6 }}


[[Gencom]]: [2 9/7; 91/90 126/125 169/168]
Optimal tunings:  
* WE: ~2 = 1198.9504{{c}}, ~3/2 = 704.6585{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.1888{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 6 8 11 0 10}}, {{val|0 -7 -9 -13 0 -10}}]
{{Optimal ET sequence|legend=0| 12, 17, 63, 80, 97c′, 114c′ }}


[[Mapping|Sval mapping]]: [{{val|1 6 8 11 10}}, {{val|0 -7 -9 -13 -10}}]
Badness (Sintel): 0.223


[[Tp tuning|POL2 generator]]: ~9/7 = 443.322
==== Sburb ====
Sburb is every other step of [[pajara]], extended to the add-23 add-41 subgroup in the same way as shrub. It can also admit a [[59/1|59th harmonic]] by setting the [[octave reduction|octave-reduced]] 413th harmonic (413/256, 827.998{{c}}) to the diminished seventh.


{{Val list|legend=1| 19, 27, 46, 111de, 157de }}
Subgroup: 2.3.25.7


[[Tp tuning #T2 tuning|RMS error]]: 1.321 cents
Comma list: 50/49, 64/63


=== Quartonic ===
{{Mapping|legend=2| 1 0 11 6 | 0 1 -4 -2 }}
: <small>''For full 13-limit extensions, see [[Orwellismic temperaments #Quartonic]].''</small>


Subgroup: 2.3.5.7.13
Optimal tunings:  
* WE: ~2 = 1197.6967{{c}}, ~3/2 = 705.6906{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.3438{{c}}


[[Comma list]]: 169/168, 325/324, 640/637
{{Optimal ET sequence|legend=0| 5, 12, 17, 39d }}


[[Gencom]]: [2 36/35; 169/168 325/324 640/637]
Badness (Sintel): 0.253


[[Gencom|Gencom mapping]]: [{{val|1 2 3 3 0 4}}, {{val|0 -11 -18 -5 0 -8}}]
===== 2.3.25.7.23 subgroup =====
Subgroup: 2.3.25.7.23


[[Mapping|Sval mapping]]: [{{val|1 2 3 3 4}}, {{val|0 -11 -18 -5 -8}}]
Comma list: 50/49, 64/63, 162/161


[[Tp tuning|POL2 generator]]: ~36/35 = 45.1821
{{Mapping|legend=2| 1 0 11 6 -5 | 0 1 -4 -2 6 }}


{{Val list|legend=1| 26, 27, 53 }}
Optimal tunings:
* WE: ~2 = 1197.5577{{c}}, ~3/2 = 705.3089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 706.7010{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.7439 cents
{{Optimal ET sequence|legend=0| 5, 12, 17 }}


=== [[Septidiasemi]] ===
Badness (Sintel): 0.419
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Septidiasemi]].''</small>


Subgroup: 2.3.5.7.13
=== Hypnosis ===
Related temperaments: [[swetismic temperaments #Hypnos|hypnos]], [[alphatricot family #Alphatrimot|alphatrimot]].
 
[[Subgroup]]: 2.3.7.11/5.13
 
[[Comma list]]: 169/168 ({{monzo| -3 -1 -1 0 2 }}), 540/539 ({{monzo| 2 3 -2 -1 0 }}), 729/728 ({{monzo| -3 6 -1 0 -1 }})
 
{{Mapping|legend=3| 1 0 -3 8 0 | 0 3 11 -13 7 }}
: mapping generators: ~2, ~13/9
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.6306{{c}}, ~13/9 = 633.8505{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/9 = 633.5546{{c}}
 
{{Optimal ET sequence|legend=1| 17, 36, 125f, 161f, 197f }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
 
=== Darian calendar ===
Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to [[36/35]]. Five of them make [[11/8]], six of them make [[32/19]], and eight of them make [[3/2]].


[[Comma list]]: 2205/2197, 2401/2400, 4096/4095
668edo does not map 36/35 consistently, with its own [[direct approximation]] being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.


[[Gencom]]: [2 15/14; 2205/2197 2401/2400 4096/4095]
Darian calendar was named by [[Eliora]] in 2023 after a certain calendar layout by the same name.


[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4}}, {{val|0 26 -37 -12 0 -3}}]
[[Subgroup]]: 2.3.35.11.19


[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4}}, {{val|0 26 -37 -12 -3}}]
[[Comma list]]: 29282/29241 ({{monzo| 1 -4 0 4 -2 }}), 42875/42768 ({{monzo| -4 -5 3 -1 0 }}), 885115/884736 ({{monzo| -15 -3 1  3 1 }})


[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
{{Mapping|legend=3| 4 5 18 13 18 | 0 8 15 5 -6 }}
: mapping generators: ~2240/1881, ~36/35


{{Val list|legend=1| 10, 151, 161, 171 }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2240/1881 = 299.9912{{c}}, ~36/35 = 50.2951{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2240/1881 = 300.0000{{c}}, ~36/35 = 50.2947{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.2002 cents
{{Optimal ET sequence|legend=1| 24, …, 524, 548, 572, 596, 620, 644, 668 }}


==== 2.3.5.7.13.17 ====
=== Hydrothermal ===
Subgroup: 2.3.5.7.13.17
Named by [[Budjarn Lambeth]] in 2024, hydrothermal tempers out 50/49, the [[jubilisma]], in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone [[mos]] is melodically interesting and flavorful. The 18-tone mos is a useful "chromatic" scale for taking subsets of.


[[Comma list]]: 833/832, 1275/1274, 2025/2023, 2205/2197
[[Subgroup]]: 2.3.7/5


[[Gencom]]: [2 15/14; 833/832 1275/1274 2025/2023 2205/2197]
[[Comma list]]: 50/49 ({{monzo| 1 0 -2 }})


[[Gencom|Gencom mapping]]: [{{val|1 -1 6 4 0 4 2}}, {{val|0 26 -37 -12 0 -3 21}}]
{{Mapping|legend=3| 2 0 1 | 0 1 0 }}
: mapping generators: ~7/5, ~3


[[Mapping|Sval mapping]]: [{{val|1 -1 6 4 4 2}}, {{val|0 26 -37 -12 -3 21}}]
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 599.6673{{c}}, ~3/2 = 702.5906{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}


[[Tp tuning|POL2 generator]]: ~15/14 = 119.297
{{Optimal ET sequence|legend=1| 2, 6, 8, 10, 12, 34, 46* }}
: <nowiki/>* Wart for 7/5


{{Val list|legend=1| 10, 151, 161, 171 }}
=== Argentic ===
Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the [[aberschisma]].


[[Tp tuning #T2 tuning|RMS error]]: 0.1867 cents
[[Subgroup]]: 2.3.7/5


=== [[Pontiac]] ===
[[Comma list]]: 5120/5103 ({{monzo| 10 -6 -1 }})
: <small>''For full 13- and 17-limit extensions, see [[Schismatic family #Pontiac]].''</small>


Subgroup: 2.3.5.7.13
{{Mapping|legend=3| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3


[[Comma list]]: 625/624 729/728 4096/4095
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7172{{c}}, ~3/2 = 702.6636{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.8166{{c}}


[[Gencom]]: [2 4/3; 625/624 729/728 4096/4095]
{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10}}, {{val|0 -1 8 -39 0 33}}]
[[Badness]] (Sintel): 0.119


[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10}}, {{val|0 -1 8 -39 33}}]
==== Edson ====
{{See also| Chromatic pairs #Edson }}


[[Tp tuning|POL2 generator]]: ~3/2 = 701.773
Edson is related to [[aberschismic family #Pele|pele]] and [[schismatic family #Andromeda|andromeda]].  


{{Val list|legend=1| 53, 118, 171, 224, 395 }}
[[Subgroup]]: 2.3.7/5.11/5.13/5


[[Tp tuning #T2 tuning|RMS error]]: 0.1525 cents
[[Comma list]]: [[196/195]] ({{monzo| 2 -1 2 0 -1 }}), [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[364/363]] ({{monzo| 2 -1 1 -2 1 }})


==== 2.3.5.7.13.17 ====
{{Mapping|legend=3| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
Subgroup: 2.3.5.7.13.17


[[Comma list]]: 625/624, 729/728, 1225/1224, 2880/2873
{{Mapping|legend=5| 1 0 -49/4 -9/4 19/4 39/4 | 0 1 29/4 5/4 -11/4 -23/4 }}


[[Gencom]]: [2 4/3; 625/624 729/728 1225/1224 2880/2873]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.4965{{c}}, ~3/2 = 703.1192{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 703.4225{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 19 0 -10 29}}, {{val|0 -1 8 -39 0 33 -60}}]
{{Optimal ET sequence|legend=1| 12, 17, 29 }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 19 -10 29}}, {{val|0 -1 8 -39 33 -60}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents


[[Tp tuning|POL2 generator]]: ~3/2 = 701.764
==== Haumea ====
{{See also| Chromatic pairs #Haumea }}


{{Val list|legend=1| 53, 118, 171, 395, 566… }}
Haumea is the add-7/5 extension of [[#Bridgetown|bridgetown]], which is in turn the add-11/5 extension of [[#Barbados|barbados]]. Other related temperamnets include [[schismatic family #Hemigari|hemigari]], [[the Archipelago #Parizekmic|parizekmic]], and [[aberschismic family #Namaka|namaka]].


[[Tp tuning #T2 tuning|RMS error]]: 0.1696 cents
[[Subgroup]]: 2.3.7/5.11/5.13/5


=== [[Mitonic]] ===
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 0 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 0 2 }}), [[847/845]] ({{monzo| 0 0 1 2 -2 }})
: <small>''For full 13- and 17-limit extensions, see [[Minortonic family #Mitonic]].''</small>


Subgroup: 2.3.5.7.13
{{Mapping|legend=3| 1 0 10 -6 -1 | 0 2 -12 9 3 }}


[[Comma list]]: 4096/4095, 4375/4374, 13720/13689
{{Mapping|legend=5| 1 0 -3/4 37/4 -27/4 -7/4 | 0 2 0 -12 9 3 }}
: mapping generators: ~2, ~26/15


[[Gencom]]: [2 10/9; 4096/4095 4375/4374 13720/13689]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7072{{c}}, ~26/15 = 951.2727{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5016{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11}}, {{val|0 17 35 -21 0 -48}}]
{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198 }}


[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11}}, {{val|0 17 35 -21 -48}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents


[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
=== Historical ===
{{Distinguish|Historical temperaments}}
{{Distinguish|History (temperament)}}


{{Val list|legend=1| 46, 125, 171, 388 }}
Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma [[4000/3993]] ([[S-expression|S10/S11]]) to set [[11/10]] equal to one-third of 4/3, and [[676/675]] ([[S-expression|S13/S15]]) to equate [[15/13]] to one-half of 4/3, and tempers out [[441/440]] ({{S|21}}) to split 11/10 into two instances of [[22/21]][[~]][[21/20]]. [[Schismatic family #Sextilifourths|Sextilifourths]] adds the [[schismatic family #Schismic|schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.


[[Tp tuning #T2 tuning|RMS error]]: 0.1442 cents
[[Subgroup]]: 2.3.7/5.11/5.13/5


==== 2.3.5.7.13.17 ====
[[Comma list]]: 364/363 ({{monzo| 2 -1 1 -2 1 }}), 441/440 ({{monzo| -3 2 2 -1 0 }}, 1001/1000 ({{monzo| -3 0 1 1 1 }})
Subgroup: 2.3.5.7.13.17


[[Comma list]]: 833/832, 1225/1224, 1701/1700, 4096/4095
{{Mapping|legend=3| 1 2 0 1 2 | 0 -6 7 2 -9 }}
: mapping generators: ~2, ~21/20


[[Gencom]]: [2 10/9; 833/832 1225/1224 1701/1700 4096/4095]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0242{{c}}, ~21/20 = 83.0177{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0144{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 -1 -3 6 0 11 5}}, {{val|0 17 35 -21 0 -48 -6}}]
{{Optimal ET sequence|legend=1| 14, 29, 101, 130, 159 }}


[[Mapping|Sval mapping]]: [{{val|1 -1 -3 6 11 5}}, {{val|0 17 35 -21 -48 -6}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents


[[Tp tuning|POL2 generator]]: ~10/9 = 182.471
=== Direct breedsmic ===
This temperament was first proposed by {{u|Royalmilktea}}. The name was established by [[Lériendil]] in 2024. Related temperaments: [[breedsmic temperaments #Hemififths|hemififths]] and [[septischismic clan #Newt|newt]].  


{{Val list|legend=1| 46, 125, 171, 388 }}
[[Subgroup]]: 2.3.49/5


[[Tp tuning #T2 tuning|RMS error]]: 0.1341 cents
[[Comma list]]: 2401/2400 ({{monzo| -5 -1 2 }})


=== [[Tertiaseptal]] ===
{{Mapping|legend=3| 1 1 3 | 0 2 1 }}
: <small>''For full 13- and 17-limit extensions, see [[Breedsmic temperaments #Tertiaseptal]].''</small>
: mapping generators: ~2, ~49/40


Subgroup: 2.3.5.7.13
[[Optimal tuning]]:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}


[[Comma list]]: 625/624, 2401/2400, 4096/4095
{{Optimal ET sequence|legend=1| 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433* }}
: <nowiki/>* Wart for 49/5


[[Gencom]]: [2 117/112; 625/624 2401/2400 4096/4095]
==== Tobago ====
{{See also| Chromatic pairs #Tobago }}


[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1}}, {{val|0 -22 5 -3 0 42}}]
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[rastmic clan #Neutral|neutral]] and [[#Barbados|barbados]].


[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1}}, {{val|0 -22 5 -3 42}}]
[[Subgroup]]: 2.3.11.13/5


[[Tp tuning|POL2 generator]]: ~117/112 = 77.173
[[Comma list]]: [[243/242]] ({{monzo| -1 5 -2 0 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})


{{Val list|legend=1| 31, 109, 140, 171, 311 }}
{{Mapping|legend=3| 2 0 -1 -2 | 0 2 5 3 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1383 cents
{{Mapping|legend=5| 2 0 -5 0 -1 -1 | 0 2 -3/2 0 5 3/2 }}
: mapping generators: ~55/39, ~26/15


==== 2.3.5.7.13.17 ====
[[Optimal tuning]]s:
Subgroup: 2.3.5.7.13.17
* [[Tp tuning|Subgroup]] [[WE]]: ~55/39 = 599.9845{{c}}, ~26/15 = 950.6637{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6776{{c}}


[[Comma list]]: 625/624, 833/832, 1225/1224, 4096/4095
{{Optimal ET sequence|legend=1| 10, 14, 24, 82, 106, 130, 154 }}


[[Gencom]]: [2 68/65; 625/624 833/832 1225/1224 4096/4095]
[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents


[[Gencom|Gencom mapping]]: [{{val|1 3 2 3 0 1 1}}, {{val|0 -22 5 -3 0 42 48}}]
==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17


[[Mapping|Sval mapping]]: [{{val|1 3 2 3 1 1}}, {{val|0 -22 5 -3 42 48}}]
[[Comma list]]: 221/220 ({{monzo| -2 0 -1 1 1 }}), 243/242 ({{monzo| -1 5 -2 0 0 }}), 289/288 ({{monzo| -5 -2 0 0 2 }})


[[Tp tuning|POL2 generator]]: ~68/65 = 77.177
{{Mapping|legend=3| 2 0 -1 -2 5 | 0 2 5 3 2 }}


{{Val list|legend=1| 31, 109f, 140, 171, 311 }}
[[Optimal tuning]]s:
* [[Tp tuning|subgroup WE]]: ~17/12 = 600.1781{{c}}, ~26/15 = 950.7656{{c}} (~15/13 = 249.2344{{c}})
* [[Tp tuning|subgroup CWE]]: ~17/12 = 600.0000{{c}}, ~26/15 = 950.6011{{c}} (~15/13 = 249.3989{{c}})


[[Tp tuning #T2 tuning|RMS error]]: 0.1367 cents
{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki/>* Wart for 13/5


=== Oquatonic ===
=== Bridgetown ===
: <small>''For full 13-limit extensions, see [[Horwell temperaments #Oquatonic]].''</small>
{{See also| Chromatic pairs #Bridgetown }}


Subgroup: 2.3.5.7.13
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is the no-7/5 restriction of [[#Haumea|haumea]] and the add-11/5 [[extension]] of [[#Barbados|barbados]].  


[[Comma list]]: 625/624, 4096/4095, 10985/10976
[[Subgroup]]: 2.3.11/5.13/5


[[Gencom]]: [40/39 105/104; 625/624 4096/4095 10985/10976]
[[Comma list]]: [[352/351]] ({{monzo| 5 -3 1 -1 }}), [[676/675]] ({{monzo| 2 -3 0 2 }})


[[Gencom|Gencom mapping]]: [{{val|28 44 65 79 0 104}}, {{val|0 1 0 -1 0 -1}}]
{{Mapping|legend=3| 1 0 -6 -1 | 0 2 9 3 }}


[[Mapping|Sval mapping]]: [{{val|28 44 65 79 104}}, {{val|0 1 0 -1 -1}}]
{{Mapping|legend=5| 1 0 7/3 0 -11/3 4/3 | 0 2 -4 0 5 -1 }}
: mapping generators: ~2, ~26/15


[[Tp tuning|POL2 generator]]: ~105/104 = 16.4240
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6281{{c}}, ~26/15 = 951.3060{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.5580{{c}}


{{Val list|legend=1| 28, 56, 84, 140, 224, 364 }}
{{Optimal ET sequence|legend=1| 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343 }}
: <nowiki/>* Wart for 11/5


[[Tp tuning #T2 tuning|RMS error]]: 0.1347 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents


== No-sevens subgroup ==
=== Seventeen-cot ===
Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the [[tendoartisma]] in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by {{u|FilterNashi}} in 2026.


=== Porkypine ===
[[Subgroup]]: 2.3.11/5.13/5
Related temperament: [[Porcupine]]


Subgroup: 2.3.5.11
[[Comma basis]]: 43940/43923 ({{monzo| 2 -1 -4 3 }}), 225000/224939 ({{monzo| 3 2 -3 -2 }})


[[Comma list]]: 55/54, 100/99
{{Mapping|legend=3| 1 1 1 1 | 0 17 4 11 }}
: mapping generators: ~2, ~169/165


[[Gencom]]: [2 10/9; 55/54, 100/99]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.993492{{c}}, ~169/165 = 41.291827{{c}}
: [[error map]]: {{val| -0.0065 -0.0004 +0.1566 -0.0104 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.000000, ~169/165 = 41.291746{{c}}
: error map: {{val| 0.0000 +0.0047 +0.1628 -0.0047 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 3 0 4}}, {{val|0 -3 -5 0 -4}}]
{{Optimal ET sequence|legend=1| 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161… }}


[[Mapping|Sval mapping]]: [{{val|1 2 3 4}}, {{val|0 -3 -5 -4}}]
[[Badness]] (Sintel): 0.080


[[Tp tuning|POL2 generator]]: ~11/10 = 164.078
=== Blackweed ===
Blackweed is a [[restriction]] of undecimal [[blackwood]] as it tempers out [[256/243]] alike but in the 2.3.11/7 subgroup. [[20edo]] is close to the optimum, which has 4\20 as the period and 420{{c}} as the generator.


{{Val list|legend=1| 7, 15, 22, 37, 73cd, 95cd, 117bcd }}
[[Subgroup]]: 2.3.11/7


[[Tp tuning #T2 tuning|RMS error]]: 2.287 cents
[[Comma list]]: 256/243 ({{monzo| 8 -5 0 }})


=== Mohaha ===
{{Mapping|legend=3| 5 8 0 | 0 0 1 }}
Related temperament: [[mohajira]], [[Meantone family #Migration|migration]]
: mapping generators: ~9/8, ~11/7


Subgroup: 2.3.5.11
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~8/7 = 238.851{{c}}, ~11/7 = 782.457{{c}}
: [[error map]]: {{val| -5.746 +8.852 -0.035 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~8/7 = 240.000{{c}}, ~11/7 = 784.967{{c}}
: error map: {{val| 0.000 +18.045 +2.475 }}


[[Comma list]]: 81/80, 121/120
{{Optimal ET sequence|legend=1| 15, 20, 35b, 55b }}


[[Gencom]]: [2 11/9; 81/80 121/120]
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 8 0 5}}]
Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain [[retraction]] of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.


[[Mapping|Sval mapping]]: [{{val|1 1 0 2}}, {{val|0 2 8 5}}]
[[Subgroup]]: 2.3.11/7.13/7


[[Tp tuning|POL2 generator]]: ~11/9 = 348.094
[[Comma list]]: 352/351 ({{monzo| 5 -3 1 -1 }}), 364/363 ({{monzo| 2 -1 -2 1 }})


{{Val list|legend=1| 7, 10, 17, 24, 31, 55, 69d, 100d, 131bd }}
{{Mapping|legend=3| 1 0 7 12 | 0 1 -4 -7 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.392 cents
{{Mapping|legend=5| 1 0 0 -19/3 2/3 17/3 | 0 1 0 11/3 -1/3 -10/3 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.3706{{c}}, ~11/7 = 703.4872{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~11/7 = 703.8328{{c}}
 
{{Optimal ET sequence|legend=1| 12, 17, 29, 75, 104 }}
 
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
 
=== Fendo ===
Fendo tempers out [[40/39]] in the 2.3.13/5 subgroup. This equates [[4/3]] with the tendo third [[13/10]]. Fendo can be viewed as an alternative interpretation of scales, chords, etc. commonly associated with [[father family #Father|father]], as it is sometimes perceived that father's characteristics tend to suggest harmonic function more than to actually serve as a temperament. It was named by [[User: VectorGraphics|Vector]] in 2025 as a contraction of ''fourth'' and ''tendo third''.
 
In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords.
 
[[5edo]] and [[13edo]] make good fendo tunings.
 
[[Subgroup]]: 2.3.13/5
 
[[Comma list]]: 40/39 ({{monzo| 3 -1 -1 }})
 
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1195.6502{{c}}, ~3/2 = 709.8204{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 710.1529{{c}}
 
{{Optimal ET sequence|legend=1| 1, 2, 3, 5, 17*, 22*, 27*, 32* }}
: <nowiki/>* Wart for 13/5
 
=== Barbados ===
Barbados is the restriction of [[#Bridgetown|bridgetown]] to the 2.3.13/5 subgroup, and tempers out [[676/675]] alone. The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.
 
[[Subgroup]]: 2.3.13/5
 
[[Comma list]]: 676/675 ({{monzo| 2 -3 2 }})
 
{{Mapping|legend=3| 1 0 -1 | 0 2 3 }}
: mapping generators: ~2, ~26/15
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9502{{c}}, ~26/15 = 951.0713{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 951.0933{{c}}
 
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 29, 53, 82, 135 }}
 
[[Badness]] (Smith): 0.002335


; Music
; Music
[http://micro.soonlabel.com/gene_ward_smith/Others/Stiltner/mohaha10ping2.mp3 Mohaha10ping2] by [[Billy Stiltner]]
* ''[[Desert Island Rain]]'' by [[Sevish]] – [https://web.archive.org/web/20201127013711/http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 play] | [https://sevish.bandcamp.com/track/desert-island-rain Bandcamp] | [https://soundcloud.com/sevish/desert-island-rain SoundCloud] | [https://www.youtube.com/watch?v=Gcgawrr2xao YouTube] – in 313edo tuned Barbados[9]
 
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and  [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.
 
[[Subgroup]]: 2.3.17/5
 
[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})
 
{{Mapping|legend=3| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}


=== Tetracot ===
{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}
: <small>''For full 11- and 13-limit extensions, see [[Tetracot family]].''</small>


Subgroup: 2.3.5.11
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.  


[[Comma list]]: 100/99, 243/242
[[Subgroup]]: 2.3.19/7


[[Gencom]]: [2 10/9; 100/99 243/242]
[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2}}, {{val|0 4 9 0 10}}]
{{Mapping|legend=3| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3


[[Mapping|Sval mapping]]: [{{val|1 1 1 2}}, {{val|0 4 9 10}}]
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}


[[Tp tuning|POL2 generator]]: ~10/9 = 175.985
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}
: <nowiki/>* Wart for 19/7


{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
[[Badness]] (Sintel): 0.082


[[Tp tuning #T2 tuning|RMS error]]: 1.182 cents
== 2.9.… subgroups ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].  


==== 2.3.5.11.13 ====
=== Baldy ===
Subgroup: 2.3.5.11.13
{{See also| Chromatic pairs #Baldy }}


[[Comma list]]: 100/99, 144/143, 243/242
Baldy is every other step of [[schismatic family #Garibaldi|garibaldi]]. It is also the add-9 extension of [[no-threes subgroup temperaments #Frostburn|frostburn]]. One of the best extension is to the 2.9.5.7.13 subgroup, mapping 13/8 to +10 whole tones, the same as the cassandra temperament.


[[Gencom]]: [2 10/9; 100/99 243/242]
[[Subgroup]]: 2.9.5.7


[[Gencom|Gencom mapping]]: [{{val|1 1 1 0 2 4}}, {{val|0 4 9 0 10 -2}}]
[[Comma list]]: 225/224, 3125/3087


[[Mapping|Sval mapping]]: [{{val|1 1 1 2 4}}, {{val|0 4 9 10 -2}}]
{{Mapping|legend=3| 1 0 15 25 | 0 1 -4 -7 }}
: mapping generators: ~2, ~9


[[Tp tuning|POL2 generator]]: ~10/9 = 176.196
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1233{{c}}, ~9/8 = 204.1913{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 204.1549{{c}}


{{Val list|legend=1| 7, 27d, 34, 41, 75d }}
{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47, 194 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.140 cents
[[Badness]] (Sintel): 0.274


=== Larry ===
==== 2.9.5.7.13 subgroup ====
Subgroup: 2.3.5.11
Subgroup: 2.9.5.7.13


[[Comma list]]: 243/242, 4000/3993
Comma list: 225/224, 325/324, 640/637


Related temperaments: [[Gravity family|gravity]], [[Gravity family #Harry|harry]]
{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}


[[Gencom]]: [2 40/27; 243/242 4000/3993]
{{Mapping|legend=4| 1 0 15 25 0 28 | 0 1/2 -4 -7 0 10 }}


[[Gencom|Gencom mapping]]: [{{val|1 5 12 0 12}}, {{val|0 -6 -17 0 -15}}]
Optimal tunings:  
* WE: ~2 = 1200.0155{{c}}, ~9/8 = 204.0931{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.0902{{c}}


[[Mapping|Sval mapping]]: [{{val|1 5 12 12}}, {{val|0 -6 -17 -15}}]
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 100, 147 }}


[[Tp tuning|POL2 generator]]: ~40/27 = 683.166
Badness (Sintel): 0.386


{{Val list|legend=1| 7, 58, 65, 137, 202 }}
RMS error: 0.5999 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.3025 cents
==== Baldanders ====
Baldanders results from taking every other generator of the [[schismatic family #Andromeda|andromeda]], mapping 11/8 to -9 whole tones.


=== Twentcufo ===
Subgroup: 2.9.5.7.11
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Undetrita]].''</small>


Subgroup: 2.3.5.11
Comma list: 100/99, 225/224, 245/242


[[Comma list]]: 8019/8000, 14641/14580
{{Mapping|legend=2| 1 0 15 25 32 | 0 1 -4 -7 -9 }}


[[Gencom]]: [2 400/363; 8019/8000 14641/14580]
Optimal tunings:  
* WE: ~2 = 1200.1917{{c}}, ~9/8 = 204.7755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.7197{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 0 -2 0 0}}, {{val|0 11 30 0 24}}]
{{Optimal ET sequence|legend=0| 6, 23de, 29, 35, 41 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -2 0}}, {{val|0 11 30 24}}]
Badness (Sintel): 0.389


[[Tp tuning|POL2 generator]]: ~400/363 = 172.8796
===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13


{{Val list|legend=1| 7, 111, 118 }}
Comma list: 100/99, 144/143, 225/224, 245/242


[[Tp tuning #T2 tuning|RMS error]]: 0.2393 cents
{{Mapping|legend=2| 1 0 15 25 32 -28 | 0 1 -4 -7 -9 10 }}


=== Emka ===
Optimal tunings:
: <small>''For full 11- and 13-limit extensions, see [[Hemimean clan #Emka]] or [[Horwell temperaments #Emkay]].''</small>
* WE: ~2 = 1199.6385{{c}}, ~9/8 = 204.3526{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/8 = 204.4234{{c}}


Subgroup: 2.3.5.11
{{Optimal ET sequence|legend=0| 6, 35, 41, 47, 88e }}


[[Comma list]]: 4000/3993, 9453125/9437184
Badness (Sintel): 0.653


[[Gencom]]: [2 11/8; 4000/3993 9453125/9437184]
=== Daemotertiaschis ===
{{See also| Schismatic family #Tertiaschis }}


[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3}}, {{val|0 -27 -8 0 1}}]
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness (hence the name – daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.


[[Mapping|Sval mapping]]: [{{val|1 14 6 3}}, {{val|0 -27 -8 1}}]
Subgroup: 2.9.5.7.33.13.17


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7778
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976


{{Val list|legend=1| 37, 50, 87, 137, 224 }}
{{Mapping|legend=2| 1 1 11 -16 13 -18 20 | 0 3 -12 26 -11 30 -22 }}
: mapping generators: ~2, ~33/20


[[Tp tuning #T2 tuning|RMS error]]: 0.1188 cents
Optimal tunings:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3019{{c}}, 33/20 = 868.1949{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, 33/20 = 867.9773{{c}}


==== 2.3.5.11.13 ====
{{Optimal ET sequence|legend=1| 47, 159, 206, 253, 459… }}
Subgroup: 2.3.5.11.13


[[Comma list]]: 625/624, 2200/2197, 4000/3993
[[Badness]] (Sintel): 0.439


[[Gencom]]: [2 11/8; 625/624 2200/2197 4000/3993]
=== A-team ===
{{See also| Dual-fifth temperaments #Dual-3 A-Team }}


[[Gencom|Gencom mapping]]: [{{val|1 14 6 0 3 6}}, {{val|0 -27 -8 0 1 -5}}]
A-team is every other step of [[gamelismic clan #Mothra|mothra]].


[[Mapping|Sval mapping]]: [{{val|1 14 6 3 6}}, {{val|0 -27 -8 1 -5}}]
[[Subgroup]]: 2.9.5.21


[[Tp tuning|POL2 generator]]: ~11/8 = 551.7753
[[Comma list]]: 81/80 ({{monzo| -4 2 -1 0 }}), 1029/1024 ({{monzo| -10 -1 0 3 }})


{{Val list|legend=1| 37, 50, 87, 137, 224 }}
{{Mapping|legend=3| 1 2 0 4 | 0 3 6 1 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.1250 cents
{{Mapping|legend=5| 1 1 0 3 | 0 3/2 6 -1/2 }}
: mapping generators: ~2, ~21/16


=== Majvam ===
[[Optimal tuning]]s:
: <small>''For full 13- and 17-limit extensions, see [[Parkleiness temperaments #Majvamic]] or [[Cataharry temperaments #Majvamoid]].''</small>
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.9303{{c}}, ~21/16 = 464.7465{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 464.5029{{c}}


Subgroup: 2.3.5.13
{{Optimal ET sequence|legend=1| 13, 18, 31 }}


[[Comma list]]: 676/675, 127401984/126953125
[[Badness]] (Sintel): 0.177


[[Gencom]]: [2 125/96; 676/675 127401984/126953125]
===== 2.9.5.21.11 subgroup =====
Subgroup: 2.9.5.21.11


[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19}}, {{val|0 -22 -7 0 0 -40}}]
Comma list: 81/80 ({{monzo| -4 2 -1 0 }}), 99/98 ({{monzo| -1 2 0 -2 1 }}), 385/384 ({{monzo| -7 -1 1 1 1 }})


[[Mapping|Sval mapping]]: [{{val|1 10 5 19}}, {{val|0 -22 -7 -40}}]
{{Mapping|legend=2| 1 2 0 4 5 | 0 3 6 1 -4 }}


[[Tp tuning|POL2 generator]]: ~125/96 = 458.988
{{Mapping|legend=4| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}


{{Val list|legend=1| 34, 149, 183, 217, 400 }}
Optimal tunings:
* Subgroup WE: ~2 = 1201.3979{{c}}, ~21/16 = 464.6020{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 464.1242{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.1171 cents
{{Optimal ET sequence|legend=0| 13, 31, 44, 75b′ }}


==== 2.3.5.13.17 ====
Badness (Sintel): 0.213
Subgroup: 2.3.5.13.17


[[Comma list]]: 676/675, 2601/2600, 24576/24565
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}


[[Gencom]]: [2 125/96; 676/675 2601/2600 24576/24565]
[[Subgroup]]: 2.9.5.11.13


[[Gencom|Gencom mapping]]: [{{val|1 10 5 0 0 19 6}}, {{val|0 -22 -7 0 0 -40 -5}}]
[[Comma list]]: 45/44, 65/64, 81/80


[[Mapping|Sval mapping]]: [{{val|1 10 5 19 6}}, {{val|0 -22 -7 -40 -5}}]
{{Mapping|legend=3| 1 0 -4 -6 10 | 0 1 2 3 -2 }}


[[Tp tuning|POL2 generator]]: ~125/96 = 458.991
{{Mapping|legend=5| 1 0 -4 0 -6 10 | 0 1/2 2 0 3 -2 }}
: mapping generators: ~2, ~9


{{Val list|legend=1| 34, 149, 183, 217, 400 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.6916{{c}}, ~9/8 = 186.1446{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 186.1740{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.1129 cents
{{Optimal ET sequence|legend=1| 6, 13, 45b′f }}


=== Photia ===
[[Badness]] (Sintel): 0.281
Related temperament: [[Schismatic family|Schismic]]


Subgroup: 2.3.5.17
[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents


[[Comma list]]: 256/255, 1458/1445
Music:
* ''[[Thundersnow]]'' by [[Sevish]] (2021)


[[Gencom]]: [2 4/3; 256/255 1458/1445]
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[Magic family #Magic|magic]].


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7}}, {{val|0 -1 8 0 0 0 -7}}]
[[Subgroup]]: 2.9.15.7


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7}}, {{val|0 -1 8 -7}}]
[[Comma list]]: 225/224 ({{monzo| -5 0 2 -1 }}), 245/243 ({{monzo| 0 -3 1 2 }})


[[Tp tuning|POL2 generator]]: ~3/2 = 701.491
{{Mapping|legend=3| 1 0 2 -1 | 0 5 3 6 }}


{{Val list|legend=1| 12, 41, 53, 65 }}
{{Mapping|legend=5| 1 0 2 -1 | 0 5/2 1/2 6 }}
: mapping generators: ~2, ~14/9


[[Tp tuning #T2 tuning|RMS error]]: 0.4842 cents
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1201.0786{{c}}, ~14/9 = 761.3878{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 760.9151{{c}}


==== 2.3.5.17.19 ====
{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 71, 112* }}
Subgroup: 2.3.5.17.19
: <nowiki/>* Wart for 15


[[Comma list]]: 171/170, 256/255, 324/323
[[Badness]] (Sintel): 0.0884


[[Gencom]]: [2 4/3; 171/170 256/255 324/323]
[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 7 3}}, {{val|0 -1 8 0 0 0 -7 3}}]
==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11


[[Mapping|Sval mapping]]: [{{val|1 2 -1 7 3}}, {{val|0 -1 8 -7 3}}]
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 }}), 225/224 ({{monzo| -5 0 2 -1 0 }}), 245/243 ({{monzo| 0 -3 1 2 0 }})


[[Tp tuning|POL2 generator]]: ~3/2 = 701.470
{{Mapping|legend=2| 1 0 2 -1 6 | 0 5 3 6 -4 }}


{{Val list|legend=1| 12, 41, 53, 65 }}
{{Mapping|legend=4| 1 0 2 -1 6 | 0 5/2 1/2 6 -4 }}
 
Optimal tunings:
* Subgroup WE: ~2 = 1200.1372{{c}}, ~14/9 = 761.4799{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 761.4017{{c}}
 
{{Optimal ET sequence|legend=0| 8, 11, 30, 41, 52, 93, 145 }}
 
Badness (Sintel): 0.166
 
RMS error: 1.226 cents


[[Tp tuning #T2 tuning|RMS error]]: 0.5374 cents
==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13


=== Nestoria ===
Comma list: 100/99 ({{monzo| 2 -2 2 0 -1 0 }}), 105/104 ({{monzo| -3 0 1 1 0 -1 }}), 144/143 ({{monzo| 4 1 0 0 -1 -1 }}), 196/195 ({{monzo| 2 0 -1 2 0 -1 }})
Related temperament: [[Schismatic family|Schismic]]


Subgroup: 2.3.5.19
{{Mapping|legend=2| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}


[[Comma list]]: 361/360, 513/512
{{Mapping|legend=4| 1 0 2 -1 6 -2 | 0 5/2 1/2 6 -4 9 }}


[[Gencom]]: [2 4/3; 361/360 513/512]
Optimal tunings:  
* Subgroup WE: ~2 = 1200.0331{{c}}, ~14/9 = 760.8754{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~14/9 = 760.8568{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 2 -1 0 0 0 0 3}}, {{val|0 -1 8 0 0 0 0 3}}]
{{Optimal ET sequence|legend=0| 11, 30, 41 }}


[[Mapping|Sval mapping]]: [{{val|1 2 -1 3}}, {{val|0 -1 8 3}}]
Badness (Sintel): 0.257


[[Tp tuning|POL2 generator]]: ~3/2 = 701.746
RMS error: 1.540 cents


{{Val list|legend=1| 12, 29, 41, 53, 118, 171 }}
=== B-team ===
B-team may be described as 23 & 41 and is every other step of [[gamelismic clan #Rodan|rodan]].


[[Tp tuning #T2 tuning|RMS error]]: 0.1763 cents
[[Subgroup]]: 2.9.15.21


== No-fives subgroup ==
[[Comma list]]: 245/243 ({{monzo| 0 -4 1 2 }}), 1029/1024 ({{monzo| -10 -1 0 3 }})


=== [[Semaphore and Godzilla|Semaphore]] ===
{{Mapping|legend=3| 1 2 0 4 | 0 3 10 1 }}
Subgroup: 2.3.7


[[Comma]]: 49/48
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.2146{{c}}, ~21/16 = 468.9174{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 468.8518{{c}}


[[Gencom]]: [2 8/7; 49/48]
{{Optimal ET sequence|legend=1| 18, 23, 41, 64, 151 }}


[[Gencom|Gencom mapping]]: [{{val|1 2 0 3}}, {{val|0 -2 0 -1}}]
[[Badness]] (Sintel): 0.155


[[Mapping|Sval mapping]]: [{{val|1 2 3}}, {{val|0 -2 -1}}]
==== 2.9.15.21.33 subgroup ====
Subgroup: 2.9.15.21.33


[[Tp tuning|POL2 generator]]: ~7/6 = 250.385
Comma list: 245/243 ({{monzo| 0 -4 1 2 0 }}), 385/384 ({{monzo| -7 -2 1 1 1 }}), 441/440 ({{monzo| -3 1 -1 2 -1 }})


{{Val list|legend=1| 5, 14, 19, 24, 67c, 91c }}
{{Mapping|legend=2| 1 2 0 4 7 | 0 3 10 1 -5 }}


[[Tp tuning# T2 tuning|RMS error]]: 2.523 cents
Optimal tunings:  
* Subgroup WE: ~2 = 1200.0553{{c}}, ~21/16 = 468.9391{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~21/16 = 468.9189{{c}}


=== Bleu ===
{{Optimal ET sequence|legend=0| 18, 23, 41, 64, 87, 151, 238b′ }}
Subgroup: 2.3.7


[[Comma]]: 17496/16807
Badness (Sintel): 0.160


[[Gencom]]: [2 54/49; 17496/16807]
=== Mabon ===
Mabon may be described as the 11 & 62 temperament in the 2.9.7 subgroup. It was derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], and named so by {{u|Dummy index}} in 2024.


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2}}, {{val|0 5 0 7}}]
[[Subgroup]]: 2.9.7


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 5 7}}]
[[Comma basis]]: 44957696/43046721


[[Tp tuning|POL2 generator]]: ~54/49 = 139.848
{{Mapping|legend=3| 1 1 -3 | 0 3 8 }}
: mapping generators: ~2, ~729/448


{{Val list| 9, 17, 43, 60c }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0000{{c}}, ~729/448 = 870.4003{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~729/448 = 871.1047{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 1.917 cents
{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62, 73b′, 157b′b′ }}


==== 2.3.7.11 ====
[[Badness]] (Sintel): 4.08
Subgroup: 2.3.7.11


[[Comma list]]: 99/98, 864/847
==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11


[[Gencom]]: [2 12/11; 99/98 864/847]
Comma basis: 896/891, 1331/1296


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3}}, {{val|0 5 0 7 4}}]
{{Mapping|legend=2| 1 1 -3 2 | 0 3 8 2 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 3}}, {{val|0 5 7 4}}]
Optimal tunings:  
* WE: ~2 = 1200.1182{{c}}, ~18/11 = 870.9904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~18/11 = 870.9234{{c}}


[[Tp tuning|POL2 generator]]: ~12/11 = 140.005
{{Optimal ET sequence|legend=0| 7d, 11, 40, 51, 62 }}


{{Val list| 9, 17, 43, 60c }}
Badness (Sintel): 4.08


[[Tp tuning #T2 tuning|RMS error]]: 1.829 cents
=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.  


==== 2.3.7.11.13 ====
It was named by [[Mike Battaglia]] in 2011, who spoke highly of this temperament, describing it as "stimulating, like a well-oiled machine".<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_96074.html Yahoo! Tuning Group | ''Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament'']</ref>
Subgroup: 2.3.7.11.13


[[Comma list]]: 78/77, 99/98, 144/143
[[Subgroup]]: 2.9.7.11


[[Gencom]]: [2 12/11; 78/77 99/98 144/143]
[[Comma list]]: 64/63, 99/98


[[Gencom|Gencom mapping]]: [{{val|1 1 0 2 3 3}}, {{val|0 5 0 7 4 6}}]
{{Mapping|legend=3| 1 0 6 13 | 0 1 -1 -3 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 3 3}}, {{val|0 5 7 4 6}}]
{{Mapping|legend=5| 1 0 0 6 13 | 0 1/2 0 -1 -3 }}
: mapping generators: ~2, ~9


[[Tp tuning|POL2 generator]]: ~12/11 = 139.990
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.2650{{c}}, ~9/8 = 213.8957{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/8 = 214.9961{{c}}


{{Val list|legend=1| 17, 43, 60c }}
{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
[[Badness]] (Sintel): 0.176


=== Archy ===
=== Apparatus ===
Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives [[dominant]]" or "no-fives [[superpyth]]". The name comes from the fact that it tempers out 64/63, the Archytas comma.
Apparatus is the restriction of [[gamelismic clan #Revelation|revelation]] to the 2.9.7.11 subgroup, and every fourth step of [[breedsmic temperaments #Quasiorwell|quasiorwell]], without prime 5.  


Subgroup: 2.3.7
[[Subgroup]]: 2.9.7.11


[[Comma]]: 64/63
[[Comma list]]: 41503/41472, 322102/321489


[[Gencom]]: [2 3/2; 64/63]
{{Mapping|legend=3| 1 -14 1 -11 | 0 19 2 16 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4}}, {{val|0 1 0 -2}}]
{{Mapping|legend=5| 1 -7 0 1 -11 | 0 19/2 0 2 16 }}
: mapping generators: ~2, ~144/77


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 -1 2}}]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0088{{c}}, ~144/77 = 1084.4377{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~144/77 = 1084.4304{{c}}


[[Tp tuning|POL2 generator]]: ~3/2 = 709.321
{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}


{{Val list|legend=1| 5, 12, 17, 22, 27, 137bc }}
[[Badness]] (Sintel): 0.199


[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.  


==== Supra ====
[[Subgroup]]: 2.9.7.11
Subgroup: 2.3.7.11


[[Comma list]]: 64/63, 99/98
[[Comma list]]: 896/891, 26411/26244
 
{{Mapping|legend=3| 1 0 -1 6 | 0 5 6 -4 }}
 
{{Mapping|legend=5| 1 0 0 -1 6 | 0 5/2 0 6 -4 }}
: mapping generators: ~2, ~14/9
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4843{{c}}, ~14/9 = 761.2079{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 761.4979{{c}}
 
{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}
 
[[Badness]] (Sintel): 0.331
 
[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
 
Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]
 
=== Joan ===
{{See also| Chromatic pairs #Joan }}
 
Joan is the restriction of [[starling temperaments #Casablanca|casablanca]] to the 2.9.7.11 subgroup, as well as every other step of [[orwell]], without prime 5.
 
[[Subgroup]]: 2.9.7.11
 
[[Comma list]]: 99/98, 9317/9216
 
{{Mapping|legend=3| 1 0 1 3 | 0 7 4 1 }}
 
{{Mapping|legend=5| 1 0 0 1 3 | 0 7/2 0 4 1 }}
: mapping generators: ~2, ~11/8
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8640{{c}}, ~11/8 = 543.0623{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/8 = 542.8446{{c}}
 
{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115b′d }}
 
[[Badness]] (Sintel): 0.407
 
[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents
 
=== Novisept ===
Named by {{u|Pailiaq}} in 2026, novisept is generated by a one-cent-flat [[9/7]], such that stacking 5 of them gives [[7/4]]. It can be formed by doubling both generator and period of [[gizzard]].
 
[[Subgroup]]: 2.9.7.13.17
 
[[Comma list]]: 442/441, 729/728, 833/832
 
{{Mapping|legend=3| 1 1 1 -1 3 | 0 6 5 13 3 }}
: mapping generators: ~2, ~9/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3044{{c}}, ~9/7 = 433.9023{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 433.8360{{c}}
 
{{Optimal ET sequence|legend=1| 11, 25f, 36, 47, 83, 130, 213g, 296g }}
 
[[Badness]] (Sintel): 0.142
 
=== Genius ===
Genius tempers out the [[Alpharabian comma]] in the 2.9.11 subgroup; it is therefore every other step of [[no-fives subgroup temperaments #Alphaxenean|alphaxenean]]. It was named by {{u|Akselai}} in 2024 after the genius in Roman religion, following the demon (daimon) in Greek mythology.
 
[[Subgroup]]: 2.9.11
 
[[Comma list]]: 131769/131072
 
{{Mapping|legend=2| 1 1 4 | 0 4 -1 }}
: mapping generators: ~2, ~16/11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3181{{c}}, ~16/11 = 650.8508{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/11 = 650.7321{{c}}
 
{{Optimal ET sequence|legend=1| 11, 24, 59, 83, 142, 225, 367e, 592b′e, 959b′eee, 1326b′b′eeee }}
 
[[Badness]] (Sintel): 0.198
 
=== Demon ===
Demon is a temperament which equates three [[11/9]]'s with [[16/9]], or equivalently three [[18/11]]'s with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by {{u|CompactStar}} while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") [[mos]], known for its lack of representation of simple temperaments. The optimal tuning for the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports this temperament.
 
[[#Mabon|Mabon]] is a strong extension of this temperament to the 2.9.7.11 subgroup.
 
[[Subgroup]]: 2.9.11
 
[[Comma list]]: 1331/1296
 
{{Mapping|legend=3| 1 1 2 | 0 3 2 }}
: mapping generators: ~2, ~18/11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.1151{{c}}, ~18/11 = 869.4724{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/11 = 869.3196{{c}}
 
{{Optimal ET sequence|legend=1| 4, 7, 11, 18, 29, 76e }}
 
[[Badness]] (Sintel): 0.332
 
== 4.3.… subgroups ==
=== Tetrameantone ===
{{Main| Tetrameantone }}
 
Named by [[Budjarn Lambeth]] in 2024, tetrameantone is the double-octave restriction of [[meantone family #Meantone|meantone]].
 
[[Subgroup]]: 4.3.5
 
[[Comma list]]: 81/80
 
{{Mapping|legend=3| 1 0 -2 | 0 1 4 }}
: mapping generators: ~4, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2402.7812{{c}}, ~3 = 1898.4361{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~3 = 1896.6512{{c}}
 
[[Optimal ET sequence]]: [[5ed4|a′5]], [[14ed4|a′14]], [[19ed4|a′19]], [[43ed4|a′43]], [[62ed4|a′62]], [[81ed4|a′81]], [[262ed4|a′262b]], [[343ed4|a′343b]]
 
[[Badness]] (Sintel): 0.0866
 
=== Tetrahanson ===
{{Main| Tetrahanson }}
 
Named by {{u|CompactStar}} in 2023, tetrahanson is the double-octave restriction of [[kleismic family #Kleismic a.k.a. hanson|hanson]].
 
[[Subgroup]]: 4.3.5
 
[[Comma list]]: 15625/15552
 
{{Mapping|legend=3| 1 -3 -2 | 0 6 5 }}
: mapping generators: ~4, ~12/5
 
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2400.3318{{c}}, ~12/5 = 1517.2163{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~12/5 = 1517.0308{{c}}
 
[[Optimal ET sequence]]: [[8ed4|a′8]], [[11ed4|a′11]], [[19ed4|a′19]], [[68ed4|a′68]], [[87ed4|a′87]], [[106ed4|a′106]], [[511ed4|a′511]], [[617ed4|a′617]], [[723ed4|a′723]], [[829ed4|a′829]], [[935ed4|a′935c]], [[1764ed4|a′1764c]]
 
[[Badness]] (Sintel): 0.155
 
=== Tetramagic ===
Named by [[Budjarn Lambeth]] in 2024, tetramagic is the double-octave restriction of [[magic family #Magic|magic]].
 
[[Subgroup]]: 4.3.5
 
[[Comma list]]: 3125/3072
 
{{Mapping|legend=3| 1 0 1 | 0 5 1 }}
: mapping generators: ~4, ~5/4
 
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2402.4898{{c}}, ~5/4 = 380.4527{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~5/4 = 380.2194{{c}}
 
[[Optimal ET sequence]]: [[6ed4|a′6]], [[13ed4|a′13]], [[19ed4|a′19]], [[63ed4|a′63]], [[82ed4|a′82]], [[101ed4|a′101]], [[120ed4|a′120]], [[221ed4|a′221c]], [[341ed4|a′341c]]
 
[[Badness]] (Sintel): 0.459


[[Gencom]]: [2 3/2; 64/63 99/98]
=== Blacktetra ===
Named by [[Budjarn Lambeth]] in 2024, blacktetra is the double-octave restriction of [[blackwood family #Blackwood|blackwood]].


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7}}, {{val|0 1 0 -2 -6}}]
[[Subgroup]]: 4.3.5


[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -2 -6}}]
[[Comma list]]: 256/243


[[Tp tuning|POL2 generator]]: ~3/2 = 707.192
{{Mapping|legend=3| 5 4 0 | 0 0 1 }}
: mapping generators: ~4, ~5


{{Val list|legend=1| 5, 12, 17, 39c, 56c }}
[[Optimal tuning]]s:
* [[WE]]: ~4/3 = 477.7018{{c}}, ~5/4 = 397.6806{{c}} (~16/15 = 80.0212{{c}})
* [[CWE]]: ~4/3 = 480.0000{{c}}, ~5/4 = 395.1255{{c}} (~16/15 = 84.7745{{c}})


[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
[[Optimal ET sequence]]: [[5ed4|a′5]], [[20ed4|a′20]], [[25ed4|a′25]], [[30ed4|a′30]]


===== Supraphon =====
[[Badness]] (Sintel): 0.748
Subgroup: 2.3.7.11.13


[[Comma list]]: 64/63, 78/77, 99/98
== 4.6.… subgroups ==
=== Meanquad ===
Meanquad is the double-octave restriction of [[meantone family #Meantone|meantone]] to the 4.6.5 subgroup. Similarly to meantone, it has a flat [[3/2]] as generator and equates [[5/4]] and [[81/64]]. It possesses [[mos scale]]s of the forms [[3L 4s (4/1-equivalent)|3L 4s<4/1>]], [[7L 3s (4/1-equivalent)|7L 3s<4/1>]], and [[7L 10s (4/1-equivalent)|7L 10s<4/1>]], which could be viewed as equivalents to the pentic, diatonic, and chromatic scales of meantone, respectively.


[[Gencom]]: [2 3/2; 64/63 78/77 99/98]
[[17ed4]] is one of the smallest tunings that [[support]] this temperament and features the decatonic scale. Like [[12edo]], it can be viewed as the boundary between meanquad and [[#quarchy|quarchy]], the 4.6.7-subgroup restriction of [[archy]]. Larger options include 24ed4 (12edo), [[31ed4]] and [[55ed4]].


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 7 9}}, {{val|0 1 0 -2 -6 -9}}]
[[Subgroup]]: 4.6.5


[[Mapping|Sval mapping]]: [{{val|1 0 6 13 18}}, {{val|0 1 -2 -6 -9}}]
[[Comma list]]: 81/80 ({{monzo| -4 4 -1 }})


[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
{{Mapping|legend=3| 1 0 -4 | 0 1 4 }}
: mapping generators: ~4, ~6


{{Val list|legend=1| 5e, 12e, 17, 22, 39c, 56c }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~4 = 2402.7812{{c}}, ~3/2 = 697.0455{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 696.6512{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
[[Optimal ET sequence]]: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[31ed4|a′31]], [[100ed4|a′100]], [[131ed4|a′131]], [[293ed4|a′293*]]
: <nowiki/>* Wart for 6


==== Suhajira ====
[[Badness]] (Sintel): 0.0146
Subgroup: 2.3.7.11


[[Comma list]]: 64/63, 243/242
==== Tetrominant ====
Subgroup: 4.6.5.7


[[Gencom]]: [2 11/9; 64/63 243/242]
Comma list: 36/35 ({{monzo| 0 2 -1 -1 }}), 64/63 ({{monzo| 4 -2 0 -1 }})


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2}}, {{val|0 2 0 -4 5}}]
{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}


[[Mapping|Sval mapping]]: [{{val|1 1 4 2}}, {{val|0 2 -4 5}}]
Optimal tunings:  
* Subgroup WE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}}
* Subgroup CWE: ~4 = 2400.0000{{c}}, ~3/2 = 699.622{{c}}


[[Tp tuning|POL2 generator]]: ~11/9 = 353.958
Optimal ET sequence: [[7ed4|a′7]], [[17ed4|a′17]], [[24ed4|a′24]], [[41ed4|a′41]], [[65ed4|a′65cd]]


{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
Badness (Sintel): 0.0974


[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
=== Quarchy ===
[[Subgroup]]: 4.6.7


===== 2.3.7.11.13 =====
[[Comma list]]: 64/63 ({{monzo| 4 -2 -1 }})
Subgroup: 2.3.7.11.13


[[Comma list]]: 64/63, 78/77, 144/143
{{Mapping|legend=3| 1 0 4 | 0 1 -2 }}
: mapping generators: ~4, ~6


[[Gencom]]: [2 11/9; 64/63 78/77 144/143]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~4 = 2393.9105{{c}}, ~3/2 = 707.5215{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~4 = 2400.0000{{c}}, ~3/2 = 709.3901{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 4 2 4}}, {{val|0 2 0 -4 5 -1}}]
Optimal ET sequence: [[7ed4|a′7]], [[10ed4|a′10]], [[17ed4|a′17]], [[27ed4|a′27]], [[44ed4|a′44]], [[159ed4|a′159d]], [[203ed4|a′203*dd]]
: <nowiki/>* Wart for 6


[[Mapping|Sval mapping]]: [{{val|1 1 4 2 4}}, {{val|0 2 -4 5 -1}}]
Badness (Sintel): 0.0134


[[Tp tuning|POL2 generator]]: ~11/9 = 353.775
== 4.9.… subgroups ==
=== Meansquared ===
Named by {{u|CompactStar}} in 2023, meansquared tempers out the double syntonic comma, [[6561/6400]], in the 4.9.25 subgroup.


{{Val list|legend=1| 7, 10, 17, 44d, 61cd, 78cd }}
[[Subgroup]]: 4.9.25


[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
[[Comma list]]: 6561/6400


=== Skwares ===
{{Mapping|legend=3| 1 0 -4 | 0 1 4 }}
Related temperament: [[Meantone family #Squares|squares]]
: mapping generators: ~4, ~9


Subgroup: 2.3.7
[[Optimal tuning]]s:  
* [[WE]]: ~4 = 2402.7812{{c}}, ~9/4 = 1394.0910{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~9/4 = 1393.3025{{c}}


[[Comma]]: 19683/19208
[[Optimal ET sequence]]: [[5ed4|a′5]], [[7ed4|a′7]], [[12ed4|a′12]], [[19ed4|a′19]], [[31ed4|a′31]], [[50ed4|a′50]], [[81ed4|a′81]], [[131ed4|a′131b′]]


[[Gencom]]: [2 9/7; 19683/19208]
[[Badness]] (Sintel): 0.173


[[Gencom|Gencom mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
=== Archsquared ===
Named by {{u|CompactStar}} in 2023, archsquared tempers out the double archytas comma, [[4096/3969]], in the 4.9.49 subgroup.


[[Mapping|Sval mapping]]: [{{val|1 3 6}}, {{val|0 -4 -9}}]
[[Subgroup]]: 4.9.49


[[Tp tuning|POL2 generator]]: ~9/7 = 425.365
[[Comma list]]: 4096/3969


{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bc, 347bc }}
{{Mapping|legend=3| 1 0 6 | 0 1 -2 }}
: mapping generators: ~4, ~9


[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
[[Optimal tuning]]s:
* [[WE]]: ~4 = 2393.9105{{c}}, ~9/4 = 1415.0430{{c}}
* [[CWE]]: ~4 = 2400.0000{{c}}, ~9/4 = 1418.7801{{c}}


==== 2.3.7.11 ====
[[Optimal ET sequence]]: [[5ed4|a′5]], [[12ed4|a′12]], [[17ed4|a′17]], [[22ed4|a′22]], [[137ed4|a′137b′d′d′]], [[159ed4|a′159b′d′d′d′]], [[181ed4|a′181b′b′d′d′d′]]
Subgroup: 2.3.7.11


[[Comma list]]: 99/98, 243/242
[[Badness]] (Sintel): 0.159


[[Gencom]]: [2 9/7; 99/98 243/242]
== 8.9.… subgroups ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, sixscared was named by [[Budjarn Lambeth]] in 2023 for the classic dad joke: "Why was six scared? Because seven ate nine."


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7}}, {{val|0 -4 0 -9 -10}}]
[[Subgroup]]: 8.9.7


[[Mapping|Sval mapping]]: [{{val|1 3 6 7}}, {{val|0 -4 -9 -10}}]
[[Comma list]]: 64/63


[[Tp tuning|POL2 generator]]: ~9/7 = 425.244
{{Mapping|legend=3| 1 0 2 | 0 1 -1 }}
: mapping generators: ~8, ~9


{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bcd }}
[[Optimal tuning]]s:
* [[WE]]: ~8 = 3590.8657{{c}}, ~9/8 = 219.0878{{c}}
* [[CWE]]: ~8 = 3600.0000{{c}}, ~9/8 = 218.7801{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
[[Optimal ET sequence]]: [[16ed8|a″16]], [[33ed8|a″33]], [[148ed8|a″148b′d]], [[181ed8|a″181b′d]], [[214ed8|a″214b′d]], [[247ed8|a″247b′b′d]]


==== 2.3.7.11.13 ====
[[Badness]] (Sintel): 0.0265
Subgroup: 2.3.7.11.13


[[Comma list]]: 78/77, 99/98, 243/242
== 2.5.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}


[[Gencom]]: [2 9/7; 78/77, 99/98, 243/242]
Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[orwellismic family #Guanyin|guanyin]] and a restriction of [[semicomma family #Orwell|orwell]]. It is defined by tempering out [[176/175]] and [[540/539]], which imply [[1728/1715]] as well as [[3388/3375]] are tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 9}}, {{val|0 -4 0 -9 -10 -15}}]
[[Subgroup]]: 2.5.7/3.11/3


[[Mapping|Sval mapping]]: [{{val|1 3 6 7 9}}, {{val|0 -4 -9 -10 -15}}]
[[Comma list]]: 176/175 ({{monzo| 4 -2 -1 1 }}), 540/539 ({{monzo| 2 1 -2 -1 }})


[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
{{Mapping|legend=3| 1 0 1 3 | 0 3 -1 5 }}


{{Val list|legend=1| 17, 48e, 65de, 82c, 147ce }}
{{Mapping|legend=5| 1 0 0 2 -2 | 0 -4/3 3 -7/3 11/3 }}
: mapping generators: ~2, ~12/7


[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2336{{c}}, ~12/7 = 929.3128{{c}} (~7/6 = 269.9208{{c}})
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 929.7608{{c}} (~7/6 = 270.2392{{c}})


==== Skwairs ====
{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
Subgroup: 2.3.7.11.13
: <nowiki/>* Wart for 7/3


[[Comma list]]: 99/98, 144/143, 243/242
[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents


[[Gencom]]: [2 9/7; 99/98, 144/143, 243/242]
==== Tridecimal guanyintet ====
Guanyintet can extend to the add-13 subgroup by the equivalences ([[12/11]])<sup>3</sup>~[[13/10]] and ([[15/14]])<sup>3</sup>~[[16/13]]. However, note that it is not supported by tridecimal orwell (31 & 53), but instead the less accurate [[semicomma family #Winston|winston]] (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. [[40edo]] remains an excellent tuning.


[[Gencom|Gencom mapping]]: [{{val|1 3 0 6 7 3}}, {{val|0 -4 0 -9 -10 2}}]
Subgroup: 2.5.7/3.11/3.13


[[Mapping|Sval mapping]]: [{{val|1 3 6 7 3}}, {{val|0 -4 -9 -10 2}}]
Comma list: 176/175 ({{monzo| 4 -2 -1 1 0 }}), 540/539 ({{monzo| 2 1 -2 -1 0 }}), 1573/1568 ({{monzo| -5 0 -2 2 1 }})


[[Tp tuning|POL2 generator]]: ~9/7 = 424.702
{{Mapping|legend=2| 1 0 1 3 1 | 0 3 -1 5 -12 }}


{{Val list|legend=1| 14, 17, 31 }}
Optimal tunings:
* Subgroup WE: ~2 = 1199.3604{{c}}, ~12/7 = 929.2866{{c}} (~7/6 = 270.0738{{c}})
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/7 = 929.7980{{c}} (~7/6 = 270.2020{{c}})


[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
{{Optimal ET sequence|legend=0| 9, 22f, 31, 40, 71, 111, 151, 262c* }}
: <nowiki/>* Wart for 7/3


=== Leapfrog ===
Badness (Sintel): 0.329
{{see also|Gentle region}}


Subgroup: 2.3.7
==== Laz ====
{{See also | Chromatic pairs #Laz }}


[[Comma list]]: 14680064/14348907
Laz is related to [[orwellismic family #Avalokita|avalokita]] as well as to [[semicomma family #Winston|winston]].


[[Gencom]]: [2 3/2; 14680064/14348907]
Subgroup: 2.5.7/3.11/3.13/3


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6}}, {{val|0 1 0 15}}]
Comma list: 144/143 ({{monzo| 4 0 0 -1 -1 }}), 176/175 ({{monzo| 4 -2 -1 1 }}), 196/195 ({{monzo| 2 -1 2 0 -1 }}


[[Mapping|Sval mapping]]: [{{val|1 0 -21}}, {{val|0 1 15}}]
{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}


[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents
{{Mapping|legend=4| 1 -3/2 0 1/2 -7/2 9/2 | 0 1/4 3 -3/4 21/4 -19/4 }}


{{Val list|legend=1| 17, 46, 63 }}
Optimal tunings:
* Subgroup WE: ~2 = 1198.4320{{c}}, ~12/7 = 929.4834{{c}} (~7/6 = 268.9486{{c}})
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/7 = 930.5797{{c}} (~7/6 = 269.4203{{c}})


[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
{{Optimal ET sequence|legend=0| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* Wart for 7/3
: † Wart for 11/3


==== 2.3.7.11 ====
RMS error: 0.8790 cents
Subgroup: 2.3.7.11


[[Comma list]]: 896/891, 1331/1323
=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}


[[Gencom]]: [2 3/2; 896/891 1331/1323]
Kryptonite is related to [[sengic family #Krypton|krypton]].


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3}}, {{val|0 1 0 15 11}}]
Subgroup: 2.5.7/3.11/3.13/3


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14}}, {{val|0 1 15 11}}]
Comma list: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})


[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}


{{Val list|legend=1| 17, 46, 63 }}
{{Mapping|legend=4| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: mapping generators: ~2, ~13/12


[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
Optimal tunings:  
* Subgroup WE: ~2 = 1197.6484{{c}}, ~13/12 = 132.1686{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~13/12 = 131.8445{{c}}


==== 2.3.7.11.13 ====
{{Optimal ET sequence|legend=0| 1, …, 8, 9 }}
Subgroup: 2.3.7.11.13


[[Comma list]]: 169/168, 352/351, 364/363
RMS error: 2.545 cents


[[Gencom]]: [2 3/2; 169/169 352/351 364/363]
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -6 -3 -1}}, {{val|0 1 0 15 11 8}}]
Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel family #Marvel|marvel]], [[magic family #Magic|magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.


[[Mapping|Sval mapping]]: [{{val|1 0 -21 -14 -9}}, {{val|0 1 15 11 8}}]
[[Subgroup]]: 2.5.9/7


[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})


{{Val list|legend=1| 17, 46, 63 }}
{{Mapping|legend=3| 1 0 5 | 0 1 -2 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
{{Mapping|legend=5| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: mapping generators: ~2, ~5


; Music
[[Optimal tuning]]s:  
* Suite for Harpsichord in A Locrian, tuning: Eb-G# in [[46edo|46EDO]] by [[User:IlL|IlL]] (in progress):
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.5971{{c}}, ~5/4 = 383.8285{{c}}
** I. Prelude
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 383.8540{{c}}
** II. Allemande
** III. Courante
** [[:File:Locrian Suite Sarabande.mp3|IV. Sarabande]] ([[:File:Locrian Suite Sarabande Score.pdf|score]], [[:File:Locrian Suite Sarabande 17edo.mp3|17EDO version]])
** [[:File:Locrian Suite Menuet.mp3|V. Menuet and Trio]]
** [[:File:Locrian Suite Gavotte.mp3|VI. Gavotte I and II]]
** [[:File:Locrian Suite Gigue.mp3|VII. Gigue]]


=== Lee ===
{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
Subgroup: 2.3.7
: <nowiki/>* Wart for 9/7


[[Comma]]: 177147/175616
[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents


[[Gencom]]: [2 81/56; 177147/175616]
==== Sulis ====
Sulis is related to [[marvel family #Minerva|minerva]] and [[würschmidt family #Septimal würschmidt|würschmidt]].


[[Gencom|Gencom mapping]]: [{{val|1 0 0 -3}}, {{val|0 3 0 11}}]
Subgroup: 2.5.9/7.11/9


[[Mapping|Sval mapping]]: [{{val|1 0 -3}}, {{val|0 3 11}}]
Comma list: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})


[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}


{{Val list|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
Optimal tunings:
* Subgroup WE: ~2 = 1200.1086{{c}}, ~5/4 = 386.5931{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~5/4 = 386.5785{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
{{Optimal ET sequence|legend=0| 3, …, 22, 25, 28, 31, 59 }}


=== [[Slendric]] ===
RMS error: 1.074 cents
Subgroup: 2.3.7


[[Comma]]: 1029/1024
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}


[[Gencom]]: [2 8/7; 1029/1024]
Magicaltet is related to [[keemic family #Keemic|keemic]], [[gamelismic clan #Superkleismic|superkleismic]], and [[magic family #Magic|magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3 0 -1}}]
[[Subgroup]]: 2.5/3.7.11


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 3 -1}}]
[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})


[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
{{Mapping|legend=3| 1 0 5 2 | 0 1 -3 2 }}


{{Val list|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
{{Mapping|legend=5| 1 0 0 5 2 | 0 -1/2 1/2 -3 2 }}
: mapping generators: ~2, ~5/3


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9868{{c}}, ~5/3 = 877.3312{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 877.3417{{c}}


==== [[Gamelismic clan #Baladic|Baladic]] ====
{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
Subgroup: 2.3.7.13
: <nowiki/>* Wart for 5/3


[[Comma list]]: 169/168, 1029/1024
[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents


[[Gencom]]: [91/64 8/7; 169/168 1029/1024]
=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}


[[Mapping|Sval mapping]]: [{{val|2 2 6 7}}, {{val|0 3 -1 1}}]
Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling family #Starling|starling]] as well as to [[starling temperaments #Myna|myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
[[Subgroup]]: 2.5/3.7/3


{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
[[Comma list]]: 126/125 ({{monzo| 1 -3 1 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
{{Mapping|legend=3| 1 0 -1 | 0 1 3 }}


==== 2.3.7.13.17 ====
{{Mapping|legend=5| 1 1/3 1/3 -2/3 | 0 -4/3 -1/3 5/3 }}
Subgroup: 2.3.7.13.17
: mapping generators: ~2, ~5/3


[[Comma list]]: 169/168, 273/272, 289/288
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7931{{c}}, ~5/3 = 888.6923{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.8022{{c}}


[[Gencom]]: [17/12 8/7; 169/168 273/272 289/288]
{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}


[[Mapping|Sval mapping]]: [{{val|2 2 6 7 7}}, {{val|0 3 -1 1 3}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents


[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}


{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226… }}
Skateboard is related to [[starling family #Thrasher|thrasher]].


[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
Subgroup: 2.5/3.7/3.11.13/9


=== Hemif ===
Comma list: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})
Related temperament: [[Breedsmic temperaments #Hemififths|hemififths]], namo


Subgroup: 2.3.7
{{Mapping|legend=2| 1 0 -1 2 2 | 0 1 3 2 -2 }}


[[Comma]]: 1605632/1594323
{{Mapping|legend=4| 1 -3/7 -3/7 -10/7 2 8/7 | 0 0 1 3 2 -2 }}


[[Gencom]]: [2 2187/1792; 1605632/1594323]
Optimal tunings:  
* Subgroup WE: ~2 = 1197.4723{{c}}, ~5/3 = 884.2917{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.6469{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1}}, {{val|0 2 0 13}}]
{{Optimal ET sequence|legend=0| 4, 11, 15, 19, 23, 42e }}


[[Mapping|Sval mapping]]: [{{val|1 1 -1}}, {{val|0 2 13}}]
RMS error: 2.396 cents


[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485
==== Greeley ====
{{See also| Chromatic pairs #Greeley }}


{{Val list|legend=1| 7, 17, 41, 58, 99 }}
Greeley is related to [[porcupine family #Opossum|opossum]] as well as to [[starling temperaments #Nusecond|nusecond]].


[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
Subgroup: 2.5/3.7/3.11/3


==== 2.3.7.11 ====
Comma list: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})
Subgroup: 2.3.7.11


[[Comma list]]: 243/242, 896/891
{{Mapping|legend=2| 1 1 2 2 | 0 -2 -6 -1 }}


[[Gencom]]: [2 11/9; 243/242 896/891]
{{Mapping|legend=4| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: mapping generators: ~2, ~11/10


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2}}, {{val|0 2 0 13 5}}]
Optimal tunings:  
* Subgroup WE: ~2 = 1200.4124{{c}}, ~11/10 = 155.8292{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~11/10 = 155.7399{{c}}


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2}}, {{val|0 2 13 5}}]
{{Optimal ET sequence|legend=0| 8, 15, 23, 54, 77, 131* }}
: <nowiki/>* Wart for 11/3


[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
RMS error: 1.034 cents


{{Val list|legend=1| 7, 17, 41, 58, 99d }}
=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 altergene{{idio}}{{clarify}} of [[no-twos subgroup temperaments #Sirius|sirius]], tempering out [[3125/3087]]. A notable tuning besides WE and CWE given below is [[CTE]]/[[POTE]] at 293.6785 [[cent]]s.


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
[[Subgroup]]: 2.5/3.7/3


==== 2.3.7.11.13 ====
[[Comma list]]: 3125/3087
Subgroup: 2.3.7.11.13


[[Comma list]]: 144/143, 243/242, 364/363
{{Mapping|legend=3| 1 0 0 | 0 3 5 }}
: mapping generators: ~2, ~25/21


[[Gencom]]: [2 11/9; 144/143 243/242 364/363]
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9983{{c}}, ~25/21 = 293.7393{{c}}
: [[error map]]: {{val| -0.002 -3.141 1.826 }}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 293.7395{{c}}
: error map: {{val| 0.000 -3.140 1.827 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 -1 2 4}}, {{val|0 2 0 13 5 -1}}]
{{Optimal ET sequence|legend=0| 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143, 192 }}
: <nowiki/>* Wart for 7/3


[[Mapping|Sval mapping]]: [{{val|1 1 -1 2 4}}, {{val|0 2 13 5 -1}}]
==== 2.5/3.7/3.13/11 subgroup ====
{{See also | Chromatic pairs #Gariberttet }}


[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].


{{Val list|legend=1| 7, 10, 17, 24, 41, 58 }}
Subgroup: 2.5/3.7/3.13/11


[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
Comma list: 275/273 ({{monzo| 0 2 -1 -1 }}), 847/845 ({{monzo| 0 -1 1 -2 }})


=== Ennea ===
{{Mapping|legend=2| 1 0 0 0 | 0 3 5 1 }}
Subgroup: 2.3.7.11


[[Comma list]]: 41503/41472, 43923/43904
{{Mapping|legend=4| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}


[[Gencom]]: [2 99/98; 41503/41472, 43923/43904]
Optimal tunings:  
* Subgroup WE: ~2 = 1199.9976{{c}}, ~13/11 = 293.6780{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~13/11 = 293.6782{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 14/9 0 25/9 31/9}}, {{val|0 2 0 2 1}}]
{{Optimal ET sequence|legend=0| 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143† }}
: <nowiki/>* Wart for 7/3
: † Wart for 13/11


[[Mapping|Sval mapping]]: [{{val|9 0 11 24}}, {{val|0 2 2 1}}]
RMS error: 0.6914 cents


[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
==== Indium ====
{{See also| Chromatic pairs #Indium }}


{{Val list|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.


[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
Subgroup: 2.5/3.7/3.11/3


=== Parapyth (Rank 3) ===
Comma list: 3025/3024 ({{monzo| -4 2 -1 2 }}), 3125/3087 ({{monzo| 0 5 -3 }})
{{see also| Pentacircle temperaments #Parapyth }}


Subgroup: 2.3.7.11
{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}


[[Comma list]]: 896/891
{{Mapping|legend=4| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: mapping generators: ~2, ~12/11


[[Gencom]]: [2 3/2 28/27; 896/891]
Optimal tunings:  
* Subgroup WE: ~2 = 1199.5499{{c}}, ~12/11 = 146.9551{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~12/11 = 146.9960{{c}}


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 }}, {{val| 0 1 0 3 -1 }}, {{val| 0 0 0 1 1 }}]
{{Optimal ET sequence|legend=0| 8, 33, 41, 49, 204*† }}
: <nowiki/>* Wart for 7/3
: † Wart for 11/3


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 }}, {{val| 0 1 0 -4 }}, {{val| 0 0 1 1 }}]
RMS error: 0.7788 cents


[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
==== Ammon ====
{{See also| Chromatic pairs #Ammon }}


{{Val list|legend=1| 17, 36, 41, 58, 63, 104 }}
Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[#Tridec|tridec]], and is related to [[porcupine family #Ammonite|ammonite]]. It is generated by ~20/13, whose octave complement, ~13/10, may be considered a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.


[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
Subgroup: 2.5/3.7/3.11/3.13/3


==== 2.3.7.11.13 ====
Comma list: 121/120 ({{monzo| -3 -1 0 2 }}), 169/168 ({{monzo| -3 0 -1 0 2 }}), 275/273 ({{monzo| 0 2 -1 1 -1 }})
Subgroup: 2.3.7.11.13


[[Comma list]]: 352/351, 364/363
{{Mapping|legend=2| 1 -3 -5 0 -1 | 0 6 10 3 5 }}


The gencom below gives [[Margo Schulter]]'s favored basis
{{Mapping|legend=4| 1 9/5 -6/5 -16/5 9/5 4/5 | 0 -24/5 6/5 26/5 -9/5 1/5 }}
: mapping generators: ~2, ~20/13


[[Gencom]]: [2 3/2 28/27; 352/351 364/363]
Optimal tunings:  
* Subgroup WE: ~2 = 1200.9125{{c}}, ~20/13 = 747.3256{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~20/13 = 746.8104{{c}}


[[Gencom]] [[mapping]]: [{{val| 1 1 0 1 4 6 }}, {{val| 0 1 0 3 -1 -4 }}, {{val| 0 0 0 1 1 1 }}]
{{Optimal ET sequence|legend=0| 8, 29, 37, 45 }}


[[Sval]] [[mapping]]: [{{val| 1 0 0 7 12 }}, {{val| 0 1 0 -4 -7 }}, {{val| 0 0 1 1 1 }}]
RMS error: 1.052 cents


[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
=== Sentry ===
{{See also | Chromatic pairs #Sentry }}


{{Val list|legend=1| 17, 41, 46, 58, 87, 104 }}
Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, tempers out [[245/243]] and shares the same generator with [[sensipent family #Sensi|sensi]]. A notable tuning besides WE and CWE given below is [[CTE]]/[[POTE]] at 440.9020 [[cent]]s.


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Subgroup]]: 2.5/3.9/7


=== Neutral ===
[[Comma list]]: 245/243 ({{monzo| 0 1 -2 }})
Neutral can be thought of as the 2.3.11 version of either [[mohajira]] or [[maqamic]], as well as suhajira and ringo. Among other things, it is the temperament optimizing the [[neutral tetrad]].


Subgroup: 2.3.11
{{Mapping|legend=3| 1 0 0 | 0 2 1 }}


[[Comma]]: 243/242
{{Mapping|legend=5| 1 0 0 0 | 0 0 2 -1 }}
: mapping generators: ~2, ~9/7


[[Gencom]]: [2 11/9; 243/242]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9983{{c}}, ~9/7 = 440.9014{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 440.9017{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2}}, {{val|0 2 0 0 5}}]
{{Optimal ET sequence|legend=1| 3, 5, 8, 11, 19, 30, 41, 49 }}


[[Mapping|Sval mapping]]: [{{val|1 1 2}}, {{val|0 2 5}}]
[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents


[[Tp tuning|POL2 generator]]: ~11/9 = 350.525
=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}


{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Kleismic family #2.3.5.13 subgroup (cata)|Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup]].


[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
[[Subgroup]]: 2.5/3.13/9


[[neutraltet7|Seven note albitonic scale]]
[[Comma list]]: 325/324 ({{monzo| -2 2 1 }})


[[neutraltet10|Ten note chromatic scale]]
{{Mapping|legend=3| 1 0 2 | 0 1 -2 }}


[[neutraltet17|Seventeen note mega chromatic scale]]
{{Mapping|legend=5| 1 -2/3 -7/3 0 0 2/3 | 0 1/2 3/2 0 0 -1 }}
: mapping generators: ~2, ~5/3


==== Namo ====
[[Optimal tuning]]s:
Subgroup: 2.3.11.13
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.1340{{c}}, ~5/3 = 882.9597{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 882.8731{{c}}


[[Comma list]]: 144/143, 243/242
{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}


[[Gencom]]: [2 11/9; 144/143 243/242]
[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents


[[Gencom|Gencom mapping]]: [{{val|1 1 0 0 2 4}}, {{val|0 2 0 0 5 -1}}]
== 2.9/5.… subgroups ==
=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}


[[Mapping|Sval mapping]]: [{{val|1 1 2 4}}, {{val|0 2 5 -1}}]
Kiribati is related to [[octagar family #Nakika|nakika]] as well as to [[tetracot family #Octacot|octacot]].


[[Tp tuning|POL2 generator]]: ~11/9 = 351.488
[[Subgroup]]: 2.9/5.7/3.11/9


{{Val list|legend=1| 7, 10, 17, 24, 41 }}
[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
{{Mapping|legend=3| 1 1 1 0 | 0 -2 3 4 }}


== No-threes subgroup ==
{{Mapping|legend=5| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
=== Llywelyn ===
: mapping generators: ~2, ~21/20
Subgroup: 2.5.7


[[Comma]]: 4194304/4117715
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.6137{{c}}, ~21/20 = 87.8636{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 87.8385{{c}}


[[Gencom]]: [2 8/7; 4194304/4117715]
{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 1 3}}, {{val|0 0 7 -1}}]
[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents


[[Mapping|Sval mapping]]: [{{val|1 1 3}}, {{val|0 7 -1}}]
=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}


[[Tp tuning|POL2 generator]]: ~8/7 = 226.910
Terrain, the 6 & 21 temperament in the 2.9/5.9/7 subgroup, tempers out the [[landscape comma]] and is related to [[minortone family #Domain|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.


{{Val list|legend=1| 16, 37 }}
[[Subgroup]]: 2.9/5.9/7


[[Tp tuning #T2 tuning|RMS error]]: 0.5391 cents
[[Comma list]]: 250047/250000 ({{monzo| -4 6 -3 }})


=== Didacus ===
{{Mapping|legend=3| 3 0 -4 | 0 1 2 }}
{{See also| Hemimean clan #Didacus }}
: mapping generators: ~63/50, ~9/5


Related temperaments: [[Chromatic pairs #Roulette|roulette]], [[Gamelismic clan #Hemithirds|hemithirds]]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~63/50 = 400.0012{{c}}, ~9/5 = 1017.5415{{c}} (~10/9 = 182.4620{{c}})
* [[Tp tuning|Subgroup]] [[CWE]]: ~63/50 = 400.0000{{c}}, ~9/5 = 1017.5388{{c}} (~10/9 = 182.4612{{c}})


Subgroup: 2.5.7
{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558, 3729, 3900, 7971 }}


[[Comma]]: 3136/3125
[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents


[[Gencom]]: [2 28/25; 3136/3125]
== 2.75.… subgroups ==
=== Archagall ===
{{See also| Fifthplus }}


[[Gencom|Gencom mapping]]: [{{val|1 0 2 2}}, {{val|0 0 2 5}}]
By tempering out the comma [[24576/24565]] in the 2.75.85 subgroup, we have three [[85/64]]'s up and one octave down as a [[75/64]] and we have two [[128/85]]'s up and one octave down as a [[17/15]] whole tone. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the ''archagallisma''. The ''MVP'' stands for ''minimum viable product'', as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high-accuracy logic" of [[superpyth]] and which is inescapably related to the [[17L 5s]] scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the 2.75.85 subgroup.


[[Mapping|Sval mapping]]: [{{val|1 2 2}}, {{val|0 2 5}}]
It is perhaps worth noting that [[83edo]] is approximately an optimal tuning for getting all of [[75/64]], [[85/64]], and the interval between them, [[17/15]], close to just, in that larger tunings compromise on tone-efficiency.


[[Tp tuning|POL2 generator]]: ~28/25 = 93.772
[[Subgroup]]: 2.75.85


{{Val list|legend=1| 6, 19, 25, 31, 37, 99, 130, 161, 353 }}
[[Comma list]]: 24576/24565 ({{monzo| 13 1 -3 }})


[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
{{Mapping|legend=3| 1 2 5 | 0 3 1 }}
: mapping generators: ~2, ~85/32


=== Rainy ===
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9692{{c}}, ~85/64 = 491.5853{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~85/64 = 491.5794{{c}}


Three generators make an [[8/7]]; five generators make a [[5/4]]. This is the no-threes version of [[tertiaseptal]].
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118…, 5939…, 7760… }}


Subgroup: 2.5.7
==== 2.75.9/7.85 subgroup ====
A fairly natural way to extend archagall is by tempering out [[2025/2023]] ([[S-expression|S15/S17]]), which equates a stack of two [[17/15]]'s with [[9/7]] without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering out S15/S16 instead (equating a stack of three [[16/15]]'s with [[17/14]]); however, [[16/15]] is not in the subgroup, so it is preferred to think of it as adding 2025/2023.


[[Comma]]: [[2100875/2097152]]
Subgroup: 2.75.9/7.85


[[Gencom]]: [2 256/245; 2100875/2097152]
Comma list: 2025/2023 ({{monzo| 2 -2 1 0 }}), 24576/24565 ({{monzo| 13 1 0 -3 }})


[[Gencom|Gencom mapping]]: [{{val|1 0 2 3 }}, {{val|0 0 5 -3}}]
{{Mapping|legend=2| 1 2 6 5 | 0 3 -4 1 }}


[[Mapping|Sval mapping]]: [{{val|1 2 3 }}, {{val|0 5 -3}}]
Optimal tunings:  
* Subgroup WE: ~2 = 1200.0241{{c}}, ~85/64 = 491.3358{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~85/64 = 491.3290{{c}}


[[Tp tuning|POL2 generator]]: ~256/245 = 77.205
{{Optimal ET sequence|legend=0| 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441… }}


{{Val list|legend=1| 31, 47, 78, 109, 140, 171, 202, 233 }}
== 2.7/3.… subgroups ==
=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0586 cents
Mothwelltri, the {{nowrap| 4 & 5 }} temperament in the 2.7/3.11 subgroup, tempers out [[99/98]] and is related to [[semicomma family #Orwell|orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.  


=== Mercy ===
[[Subgroup]]: 2.7/3.11
{{See also| Quince clan #Mercy }}


Two generators make an [[8/7]]; seven generators make an [[8/5]]. Mercy can be thought of as a way to conceptualize the 2.5.7.13.17.19 subgroup of [[31edo]], and is the no-threes or elevens version of [[miracle]].
[[Comma list]]: 99/98 ({{monzo| -1 -2 1 }})


Subgroup: 2.5.7
{{Mapping|legend=3| 1 0 1 | 0 1 2 }}


[[Comma list]]: 823543/819200
{{Mapping|legend=5| 1 0 0 0 1 | 0 -1/2 0 1/2 2 }}
: mapping generators: ~2, ~7/3


[[Gencom]]: [2 2744/2560; 823543/819200]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.3175{{c}}, ~7/6 = 273.2462{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 273.4280{{c}}


[[Gencom|Gencom mapping]]: [{{val| 1 0 3 3 }}, {{val| 0 0 -7 -2 }}]
{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}


[[Sval]] [[mapping]]: [{{val| 1 3 3 }}, {{val| 0 -7 -2 }}]
[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents


[[Tp tuning|POL2 generator]]: ~343/320 = 116.291
== 2.7/5.… subgroups ==
=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}


{{Val list|legend=1| 10, 21, 31, 134, 165, 196, 227, 485d, 712d, 1197dd }}
Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].


==== 2.5.7.13 ====
[[Subgroup]]: 2.7/5.11/5.13/5
Subgroup: 2.5.7.13


[[Comma list]]: 343/338, 640/637
[[Comma list]]: 847/845 ({{monzo| 0 1 2 -2 }}), 1001/1000 ({{monzo| -3 1 1 1 }})


[[Gencom]]: [2 14/13; 343/338 640/637]
{{Mapping|legend=3| 1 -2 3 2 | 0 4 -3 -1 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4}}, {{val|0 0 -7 -2 -3}}]
{{Mapping|legend=5| 1 0 -3/4 -11/4 9/4 5/4 | 0 0 0 4 -3 -1 }}
: mapping generators: ~2, ~20/13


[[Mapping|Sval mapping]]: [{{val|1 3 3 4}}, {{val|0 -7 -2 -3}}]
[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[WE]]: ~2 = 1200.0343{{c}}, ~20/13 = 745.4658{{c}}
* [[Tp tuning|subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~20/13 = 745.4431{{c}}


[[Tp tuning|POL2 generator]]: ~14/13 = 116.094
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66 }}


{{Val list|legend=1| 10, 21, 31}}
[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents


==== 2.5.7.13.17 ====
==== Naiadec ====
Subgroup: 2.5.7.13.17
Subgroup: 2.7/5.11/5.13/5.17/5


[[Comma list]]: 170/169, 224/221, 640/637
Comma list: 170/169 ({{monzo| 1 0 0 -2 1 }}), 221/220 ({{monzo| -2 0 -1 1 1 }}), 847/845 ({{monzo| 0 1 2 -2 0 }})


[[Gencom]]: [2 14/13; 170/169 224/221 640/637]
{{Mapping|legend=2| 1 -2 3 2 3 | 0 4 -3 -1 -2 }}


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4}}, {{val|0 0 -7 -2 -3 1}}]
{{Mapping|legend=4| 1 0 -3/4 -11/4 9/4 5/4 9/4 | 0 0 0 4 -3 -1 -2 }}
: mapping generators: ~2, ~20/13


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4}}, {{val|0 -7 -2 -3 1}}]
Optimal tunings:  
* Subgroup WE: ~2 = 1200.4068{{c}}, ~20/13 = 745.3706{{c}}
* Subgroup CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.0940{{c}}


[[Tp tuning|POL2 generator]]: ~14/13 = 115.769
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95*, 124* }}
: <nowiki/>* Wart for 17/5


{{Val list|legend=1| 10, 21, 31}}
RMS error: 0.7521 cents


==== 2.5.7.13.17.19 ====
== 2.25/7.… subgroups ==
Subgroup: 2.5.7.13.17.19
=== Supramine ===
Supramine extends [[#Supramin|supramin]] and approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.


[[Comma list]]: 170/169, 343/338, 640/637, 16384/16055
[[Subgroup]]: 2.25/7.17/7.19/7.23/7


[[Gencom]]: [2 14/13; 170/169 343/338 640/637 16384/16055]
[[Comma list]]: 323/322 ({{monzo| -1 0 1 1 -1 }}), 392/391 ({{monzo| 3 0 -1 0 -1 }}), 476/475 ({{monzo| 2 -1 1 -1 0 }})


[[Gencom|Gencom mapping]]: [{{val|1 0 3 3 4 4 3}}, {{val|0 0 -7 -2 -3 1 13}}]
{{Mapping|legend=3| 1 -2 0 4 3 | 0 3 1 -2 -1 }}
: mapping generators: ~2, ~17/14


[[Mapping|Sval mapping]]: [{{val|1 3 3 4 4 3}}, {{val|0 -7 -2 -3 1 13}}]
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.7558{{c}}, ~17/14 = 335.4278{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.4885{{c}}


[[Tp tuning|POL2 generator]]: ~14/13 = 115.716
{{Optimal ET sequence|legend=1| 7, 18, 25, 68, 93 }}


{{Val list|legend=1| 10, 21, 31, 52f}}
[[Badness]] (Sintel): 0.053


== 2.9.7.11 subgroup ==
== 2.11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}


=== Machine ===
Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup and tempers out [[2200/2197]]. It restricts [[#Tridec|tridec]]. The tonic and the first two generator steps make a [[petrmic triad]], hence the name.  
Subgroup: 2.9.7.11


Commas: 64/63, 99/98
[[Subgroup]]: 2.11/5.13/5


[[Gencom]]: [2 8/7; 64/63 99/98]
[[Comma list]]: 2200/2197 ({{monzo| 3 1 -3 }})


[[Gencom|Gencom mapping]]: [{{val|1 3/2 0 3 4}}, {{val|0 1/2 0 -1 -3}}]
{{Mapping|legend=3| 1 0 1 | 0 3 1 }}


[[Mapping|Sval mapping]]: [{{val|1 0 6 13}}, {{val|0 1 -1 -3}}]
{{Mapping|legend=5| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: mapping generators: ~2, ~13/10


[[Tp tuning|POL2 generator]]: ~8/7 = 214.384
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.9572{{c}}, ~13/10 = 454.9958{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~13/10 = 455.0039{{c}}


{{Val list|legend=1| 5, 6, 11, 17, 28 }}
{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 153, 182, 211, 240, 269, 509, 778…, 1287… }}


[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents


=== Mechanism ===
== Other 2-repeating subgroups ==
Subgroup: 2.9.7.11
=== Supramin ===
Supramin restricts [[#Supramine|supramine]] and is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is [[25edo]], which provides an accurate yet tone-efficient tuning of this temperament. It was named by {{u|Overthink}} in 2026 after the fact that the generator is a [[17/14]] supraminor third, two of which reach [[28/19]]. It is related to [[Mabila family #Cohemimabila|cohemimabila]].


[[Comma list]]: 896/891, 26411/26244
[[Subgroup]]: 2.17/7.19/7


[[Gencom]]: [2 9/7; 896/891 26411/26244]
[[Comma list]]: 5491/5488 ({{monzo| -4 2 1 }})


[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 5 2}}, {{val|0 -5/2 0 -6 4}}]
{{Mapping|legend=3| 1 0 4 | 0 1 -2 }}
: mapping generators: ~2, ~17/7


[[Mapping|Sval mapping]]: [{{val|1 5 5 2}}, {{val|0 -5 -6 4}}]
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1200.0220{{c}}, ~17/14 = 335.7933{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~17/14 = 335.7852{{c}}


[[Tp tuning|POL2 generator]]: ~9/7 = 438.465
{{Optimal ET sequence|legend=1| 7, 18, 25, 118, 143, 168, 193, 218, 461, 679, 1576, 2255… }}


{{Val list|legend=1| 8, 11, 30, 41, 52 }}
[[Badness]] (Sintel): 0.005


[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents
== 3/2.5/2.… subgroups ==
{{Main| Half-prime subgroup }}


=== Apparatus ===
=== Hemihemi ===
Subgroup: 2.9.7.11
Named by {{u|CompactStar}} in 2023, hemihemi tempers out 10976/10935, the [[hemimage comma]], in the 3/2.5/2.7/2 subgroup.  


[[Comma list]]: 41503/41472, 322102/321489
[[Subgroup]]: 3/2.5/2.7/2


[[Gencom]]: [2 77/72; 41503/41472 322102/321489]
[[Comma list]]: 10976/10935 ({{monzo| -7 -1 3 }})


[[Gencom|Gencom mapping]]: [{{val|1 5/2 0 3 5}}, {{val|0 -19/2 0 -2 -16}}]
{{Mapping|legend=3| 1 2 3 | 0 3 1 }}
: mapping generators: ~3/2, ~28/27


[[Mapping|Sval mapping]]: [{{val|1 5 3 5}}, {{val|0 -19 -2 -16}}]
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 702.6069{{c}}, ~28/27 = 60.4665{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~28/27 = 60.8011{{c}}


[[Tp tuning|POL2 generator]]: ~77/72 = 115.570
[[Optimal ET sequence]]: [[11edf|*11]], [[12edf|*12]], [[23edf|*23]], [[35edf|*35]], [[58edf|*58]], [[93edf|*93]], [[151edf|*151†]], [[244edf|*244†]]
: <nowiki/>* Wart for 3/2
: † Wart for 7/2


{{Val list|legend=1| 10, 21, 31, 52, 83, 135, 353, 488, 623 }}
=== Halftone ===
{{Main| Halftone }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0673 cents
Named by {{u|CompactStar}} in 2023, halftone tempers out 9604/9375 in the 3/2.5/2.7/2 subgroup.  


== 2.9.15.7 subgroup ==
[[Subgroup]]: 3/2.5/2.7/2


=== Stacks (aka 2magic) ===
[[Comma list]]: 9604/9375 ({{monzo| -1 -5 4 }})
Subgroup: 2.9.15.7


[[Comma list]]: 225/224, 245/243
{{Mapping|legend=3| 1 3 4 | 0 -4 -5 }}
: mapping generators: ~3/2, ~15/14


[[Gencom]]: [2 9/7; 225/224 245/243]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 702.6568{{c}}, ~15/14 = 128.2849{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~15/14 = 128.8183{{c}}


[[Gencom|Gencom mapping]]: [{{val|1 5/2 5/2 5}}, {{val|0 -5/2 -1/2 -6}}]
[[Optimal ET sequence]]: [[5edf|*5]], [[11edf|*11]], [[27edf|*27]], [[38edf|*38]]
: <nowiki/>* Wart for 3/2


[[Mapping|Sval mapping]]: [{{val|1 0 2 -1}}, {{val|0 5 3 6}}]
==== 3/2.5/2.7/2.11/2 subgroup ====
Subgroup: 3/2.5/2.7/2.11/2


[[Tp tuning|POL2 generator]]: ~9/7 = 439.296
Comma list: 1232/1215 ({{monzo| -5 -1 1 1 }}), 9604/9375 ({{monzo| -1 -5 4 0 }})


{{Val list|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}
{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents
Optimal tunings:  
* WE: ~3/2 = 704.8651{{c}}, ~15/14 = 130.9114{{c}}
* CWE: ~3/2 = 701.9550{{c}}, ~15/14 = 129.6376{{c}}


==== 2.9.15.7.11 ====
Optimal ET sequence: [[5edf|*5]], [[11edf|*11]], [[16edf|*16]], [[27edf|*27…]], [[43edf|*43…]], [[70edf|*70…]]
Subgroup: 2.9.15.7.11
: <nowiki/>* Wart for 3/2


Comma list: 100/99, 225/224, 245/243
==== 3/2.5/2.7/2.11/2.13/2 subgroup ====
Subgroup: 3/2.5/2.7/2.11/2.13/2


Gencom: [2 9/7; 100/99 225/224 245/243]
Comma list: 275/273 ({{monzo| -1 2 -1 1 -1 }}), 1232/1215 ({{monzo| -5 -1 1 1 0 }}), 1323/1300 ({{monzo| 3 -2 2 0 -1 }})


Gencom mapping: [{{val|1 5/2 5/2 5 2}}, {{val|0 -5/2 -1/2 -6 4}}]
{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}


Sval mapping: [{{val|1 0 2 -1 6}}, {{val|0 5 3 6 -4}}]
Optimal tunings:  
* WE: ~3/2 = 702.9683{{c}}, ~15/14 = 130.0424{{c}}
* CWE: ~3/2 = 701.9550{{c}}, ~15/14 = 129.6034{{c}}


POL2 generator: ~9/7 = 438.607
Optimal ET sequence: [[5edf|*5]], [[11edf|*11]], [[16edf|*16]], [[27edf|*27†]]
: <nowiki/>* Wart for 3/2
: † Wart for 11/2


Vals: {{Val list| 8, 11, 30, 41, 52, 93, 145, 342bce }}
=== Semiwolf ===
Semiwolf provides an interpretation for the {{MOS scalesig|link=true|3L 2s<3/2>}} [[mos scale]]. The name ''semiwolf'' comes from two [[~]][[7/6]] generators approximating a [[27/20]] wolf fourth. This is further extended to the 11-limit in two interpretations: semilupine, where 2 major 2-mossteps (LL) equal 11/9, and hemilycan. where 1 major and 2 minor 2-mossteps (sLs) equal 11/9. Basic [[8edf]] fits both extensions.


RMS error: 1.226 cents
[[Subgroup]]: 3/2.5/2.7/4


==== 2.9.15.7.11.13 ====
[[Comma list]]: 245/243 ({{monzo| -5 1 2 }})
Subgroup: 2.9.15.7.11.13


Comma list: 100/99, 105/104, 144/143, 196/195
{{Mapping|legend=3| 1 1 2 | 0 2 -1 }}
: mapping generators: ~3/2, ~9/7


Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 703.7414{{c}}, ~9/7 = 440.9014{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~9/7 = 440.8307{{c}}


Gencom mapping: [{{val|1 5/2 5/2 5 2 7}}, {{val|0 -5/2 -1/2 -6 4 -9}}]
[[Optimal ET sequence]]: [[8edf|*8]], [[43edf|*43]], [[51edf|*51]], [[59edf|*59]], [[67edf|*67]], [[75edf|*75…]], [[83edf|*83…]]
: <nowiki/>* Wart for 3/2


Sval mapping: [{{val|1 0 2 -1 6 -2}}, {{val|0 5 3 6 -4 9}}]
==== Hemilycan ====
Subgroup: 3/2.5/2.7/4.11/4


POL2 generator: ~9/7 = 438.977
Comma list: 245/243 ({{monzo| -5 1 2 0 }}), 441/440 ({{monzo| 2 -1 2 -1 }})


Vals: {{Val list| 11, 30, 41, 153cdef, 194cdef, 235cdef }}
{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}


RMS error: 1.540 cents
Optimal tunings:  
* Subgroup WE: ~3/2 = 703.2870{{c}}, ~9/7 = 441.1967{{c}}
* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 440.4501{{c}}


== 2.9.21 subgroup ==
Optimal ET sequence: [[8edf|*8]], [[27edf|*28]], [[35edf|*35]], [[43edf|*43]], [[51edf|*51]], [[110edf|110…]]
: <nowiki/>* Wart for 3/2


=== A-team ===
==== Semilupine ====
Subgroup: 2.9.21
Subgroup: 3/2.5/2.7/4.11/4


[[Comma]]: 1029/1024
Comma list: 100/99 ({{monzo| -3 2 0 -1 }}), 245/243 ({{monzo| -5 1 2 0 }})


[[Gencom]]: [2 21/16; 1029/1024]
{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}


[[Gencom|Gencom mapping]]: [{{val|1 1 0 3}}, {{val|0 3/2 0 -1/2}}]
Optimal tunings:  
* Subgroup WE: ~3/2 = 704.6266{{c}}, ~9/7 = 439.2433{{c}}
* Subgroup CWE: ~3/2 = 701.9550{{c}}, ~9/7 = 438.6989{{c}}


[[Mapping|Sval mapping]]: [{{val|1 2 4}}, {{val|0 3 1}}]
Optimal ET sequence: [[5edf|*5]], [[8edf|*8]], [[45edf|*45…]], [[53edf|*53…]], [[61edf|*61…]], [[69edf|*69…]], [[77edf|*77…]]
: <nowiki/>* Wart for 3/2


[[Tp tuning|POL2 generator]]: ~21/16 = 467.375
== 3/2.5/4.… subgroups ==
=== Ambulatory ===
Ambulatory tempers out [[686/675]] in the 3/2.5/4.7/4 subgroup and is generated by a sharp [[~]][[15/14]]. This implies a stack of two generators represents [[7/6]], and a stack of three generators represents [[5/4]].  


{{Val list|legend=1| 5, 13, 18, 41, 59, 77, 95 }}
Ambulatory can be considered the 3/2.5/4.7/4 analog of [[no-twos subgroup temperaments #BPS|BPS]] or [[meantone family #Meantone|meantone]], using [[5:6:7]] as its fundamental consonant chord in the place of [[3:5:7]] or of [[4:5:6]]. Ambulatory is actually similar to meantone in several ways:
* They are both of low [[badness]].{{dubious}}
* In most [[optimal tuning]]s, meantone tunes [[3/2]] flat and [[5/4]] slightly sharp, while ambulatory tunes [[7/5]] flat and [[6/5]] slightly sharp.
* Meantone[7] has [[5/4]] and [[6/5]] share a scale degree. Ambulatory[5] has [[6/5]] and [[7/6]] share a scale degree.
* Meantone[7] is a tempering of the 7-note [[AGS]](5/4, 6/5) (Ptolemy's intense diatonic) with the large and medium steps equated. Ambulatory[5] is a tempering of the 5-note AGS(6/5, 7/6) (assuming an equivalence of 3/2) with the medium and small steps equated.


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
Ambulatory was first documented by {{u|CompactStar}} who originally proposed the name ''emergency'', which refers to the North American  emergency telephone number 911, because [[11edf|9\11edf]] is a possible generator. This was deemed as too dark so the name ''ambulatory''" was suggested by {{u|Frostburn}} to more subtly reference the same thing.


== 2.11.13.17.19 subgroup ==
[[Subgroup]]: 3/2.5/4.7/4


=== Yamablu ===
[[Comma list]]: 686/675 ({{monzo| -3 -2 3 }})
Yamablu, with a generator of ~17/13, is named for it's tempering of the yama comma (209/208) and the blume comma (2057/2048), which also implies the blumeyer comma (2432/2431). The [[Kite's Method of Naming Rank-2 Scales using Mode Numbers|13th Yamablu[13]]] scale is a linear-temperament version of [[Gjaeck]].


Subgroup: 2.11.13.17.19
{{Mapping|legend=3| 1 0 1 | 0 3 2 }}]
: mapping generators: ~3/2, ~15/14


[[Comma list]]: 209/208, 2057/2048, 83521/83486
[[Optimal tuning]]s:  
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 701.8452{{c}}, ~15/14 = 129.7976{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~15/14 = 129.5324{{c}}


[[Mapping|Sval mapping]]: [{{val|1 5 1 1 0}}, {{val|0 -4 7 8 11}}]
[[Optimal ET sequence]]: [[5edf|*5]], [[11edf|*11]], [[27edf|*27]], [[38edf|*38]]
: <nowiki/>* Wart for 3/2


[[Tp tuning|POL2 generator]]: ~17/13 = 462.9606
=== Poseidon ===
{{Todo|inline=1|rename|resolve temperament name conflict}}


{{Val list|legend=1| 13, 44, 57, 70}}
[[Subgroup]]: 3/2.5/4.11/8


[[Tp tuning #T2 tuning|RMS error]]: 0.4898 cents
[[Comma list]]: 121/120 ({{monzo| -1 -1 2 }})


== Miscellaneous subgroup temperaments ==
{{Mapping|legend=3| 1 1 1 | 0 -2 -1 }}]
: mapping generators: ~3/2, ~12/11


=== Historical ===
[[Optimal tuning]]s:
Subgroup: 2.3.7/5.11/5.13/5
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 701.8452{{c}}, ~12/11 = 157.8337{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~12/11 = 157.8574{{c}}


[[Comma list]]: 364/363, 441/440, 1001/1000
[[Optimal ET sequence]]: [[4edf|*4]], [[5edf|*5]], [[9edf|*9]], [[22edf|*22]], [[31edf|*31]], [[40edf|*40]]
: <nowiki/>* Wart for 3/2


[[Mapping|Sval mapping]]: [{{val|1 2 0 1 2}}, {{val|0 -6 7 2 -9}}]
== Other 3/2-repeating subgroups ==
=== Auk ===
Auk tempers out the comma [[87808/85293]] in the 3/2.7.13 subgroup, equating three instances of [[7/1]] (fifth-reduced to [[112/81]]) with [[13/1]] (fifth-reduced to 832/729). The generator when reduced is a superfourth in the neighborhood of 550 cents, representing 112/81. It possesses nontrivial mos of the families {{MOS scalesig|1L 3s<3/2>}} ("neptunian"), {{MOS scalesig|4L 1s<3/2>}}, {{MOS scalesig|5L 4s<3/2>}}, and {{MOS scalesig|9L 5s<3/2>}}. [[9edf]] is a particularly good equal temperament that [[support]]s it. It contains tempered versions of the chords 1–832/729–112/81 (fifth-reduction of 1:7:13) and 1–63/52–112/81 (fifth-reduction of 1/(13:7:1)).


[[Tp tuning|POL2 generator]]: ~21/20 = 83.016
[[Subgroup]]: 3/2.7.13


{{Val list|legend=1| 14, 29, 72, 101, 130, 159 }}
[[Comma list]]: 87808/85293


[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents
{{Mapping|legend=3| 1 0 -8 | 0 1 3 }}
: mapping generators: ~3/2, ~7


=== Hypnosis ===
[[Optimal tuning]]s:
Related temperament: [[Swetismic temperaments #Hypnos|hypnos]], [[Hemifamity temperaments #Tricot|tricot]]
* [[Tp tuning|Subgroup]] [[WE]]: ~3/2 = 706.5197{{c}}, ~112/81 = 538.8834{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~3/2 = 701.9550{{c}}, ~112/81 = 543.8799{{c}}


Subgroup: 2.3.7.11/5.13
[[Optimal ET sequence]]: [[4edf|*4]], [[13edf|*13]], [[17edf|*17df]], [[21edf|*21df]], [[38edf|*38ddff]], [[59edf|*59ddffff]]
: <nowiki/>* Wart for 3/2


[[Comma list]]: 169/168, 540/539, 729/728
=== Doubleton ===
[[Subgroup]]: 3/2.7.13


[[Mapping|Sval mapping]]: [{{val|1 0 -3 8 0}}, {{val|0 3 11 -13 7}}]
[[Comma list]]: 1352/1323 ({{monzo| -3 -2 2 }})


[[Tp tuning|POL2 generator]]: ~13/9 = 633.518
{{Mapping|legend=3| 2 0 3 | 0 1 1 }}
: mapping generators: ~26/21, ~7


{{Val list|legend=1| 17, 36, 118e, 125e, 161e, 197e }}
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~26/21 = 352.6518{{c}}, ~112/81 = 552.6017{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~26/21 = 350.9775{{c}}, ~112/81 = 565.7161{{c}}


[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents
[[Optimal ET sequence]]: [[4edf|*4]], [[10edf|*10]], [[14edf|*14f]], [[46edf|*46dfff]]
: <nowiki/>* Wart for 3/2


=== Oceanfront ===
== 5/2-repeating subgroups ==
Subgroup: 2.3.7.13/5
=== Hyperion ===
[[Subgroup]]: 5/2.7.11


Related temperament: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]
[[Comma list]]: 341796875/329832448 ({{monzo| 11 1 -5 }})


[[Comma list]]: 64/63, 91/90
{{Mapping|legend=3| 1 -1 2 | 0 5 1 }}
: mapping generators: ~5/2, ~44/25


[[Mapping|Sval mapping]]: [{{val|1 0 6 -5}}, {{val|0 1 -2 4}}]
[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~5/2 = 1582.2958{{c}}, ~44/25 = 990.1327{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~5/2 = 1586.3137{{c}}, ~44/25 = 991.6570{{c}}


[[Tp tuning|POL2 generator]]: ~3/2 = 713.910
[[Optimal ET sequence]]: [[8ed5/2|*8]], [[83ed5/2|*83…]], [[91ed5/2|*91…]], …, [[155ed5/2|*155…]], [[163ed5/2|*163…]]
: <nowiki/>* Wart for 5/2


{{Val list|legend=1| 5, 22, 27, 32, 37 }}
== See also ==
* [[Dual-fifth temperaments]]
* [[Irrational-subgroup temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments


[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents
== References ==
<references/>


[[Category:Regular temperament theory]]
[[Category:Catalogs of subgroup temperaments]]
[[Category:Temperament clan]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Subgroup]]

Latest revision as of 10:26, 17 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

A subgroup temperament is a regular temperament defined on a just intonation subgroup that is not a full p-limit group.

For temperaments that omit various prime harmonics, see:

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on Chromatic pairs.

This page uses an extended wart notation where squares and cubes of a prime basis element are denoted by ′ and ″, respectively, to reduce the use of temporary warts.

2.3.… subgroups

Shrub

Named by Vector in 2026, shrub is a restriction of diaschismic which omits the tritone to produce a diatonic scale. True to its name, it generates a shrubmajor[idiosyncratic term] third (~425 ¢) in quarter-comma tuning. It has an equal-temperament join of 12 & 17.

It can also admit a 41st harmonic by tempering out 82/81 (same mapping as reversed meantone). However, its tuning tends to be flat, and with the rest of the harmonics tending sharp, this will increase the overall error.

2.3.25 subgroup

Subgroup: 2.3.25

Comma list: 2048/2025

Subgroup-val mapping: [⟨1 0 11], ⟨0 1 -4]]

mapping generators: ~2, ~3

Optimal tunings:

  • WE: ~2 = 1198.8214 ¢, ~3/2 = 704.2059 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.9585 ¢

Optimal ET sequence: 12, 17, 46, 63, 80, 143c′

Badness (Sintel): 0.234

2.3.25.23 subgroup

Subgroup: 2.3.25.23

Comma list: 576/575, 736/729

Subgroup-val mapping: [⟨1 0 11 -5], ⟨0 1 -4 6]]

Optimal tunings:

  • WE: ~2 = 1198.9504 ¢, ~3/2 = 704.6585 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 705.1888 ¢

Optimal ET sequence: 12, 17, 63, 80, 97c′, 114c′

Badness (Sintel): 0.223

Sburb

Sburb is every other step of pajara, extended to the add-23 add-41 subgroup in the same way as shrub. It can also admit a 59th harmonic by setting the octave-reduced 413th harmonic (413/256, 827.998 ¢) to the diminished seventh.

Subgroup: 2.3.25.7

Comma list: 50/49, 64/63

Subgroup-val mapping: [⟨1 0 11 6], ⟨0 1 -4 -2]]

Optimal tunings:

  • WE: ~2 = 1197.6967 ¢, ~3/2 = 705.6906 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 707.3438 ¢

Optimal ET sequence: 5, 12, 17, 39d

Badness (Sintel): 0.253

2.3.25.7.23 subgroup

Subgroup: 2.3.25.7.23

Comma list: 50/49, 64/63, 162/161

Subgroup-val mapping: [⟨1 0 11 6 -5], ⟨0 1 -4 -2 6]]

Optimal tunings:

  • WE: ~2 = 1197.5577 ¢, ~3/2 = 705.3089 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 706.7010 ¢

Optimal ET sequence: 5, 12, 17

Badness (Sintel): 0.419

Hypnosis

Related temperaments: hypnos, alphatrimot.

Subgroup: 2.3.7.11/5.13

Comma list: 169/168 ([-3 -1 -1 0 2⟩), 540/539 ([2 3 -2 -1 0⟩), 729/728 ([-3 6 -1 0 -1⟩)

Subgroup-val mapping: [⟨1 0 -3 8 0], ⟨0 3 11 -13 7]]

mapping generators: ~2, ~13/9

Optimal tunings:

  • Subgroup WE: ~2 = 1200.6306 ¢, ~13/9 = 633.8505 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/9 = 633.5546 ¢

Optimal ET sequence: 17, 36, 125f, 161f, 197f

RMS error: 0.5379 cents

Darian calendar

Darian calendar is described as the 24 & 668 temperament in the 2.3.35.11.19 subgroup. The generator is close to 36/35. Five of them make 11/8, six of them make 32/19, and eight of them make 3/2.

668edo does not map 36/35 consistently, with its own direct approximation being 27 steps while the direct approximations of its constituent odd harmonics do not sum to that same amount: 3/2, 8/5, and 8/7 are 391, 453, and 129 steps, respectively, and 391 + 391 + 453 + 129 - 668 - 668 = 28, ≠ 27.

Darian calendar was named by Eliora in 2023 after a certain calendar layout by the same name.

Subgroup: 2.3.35.11.19

Comma list: 29282/29241 ([1 -4 0 4 -2⟩), 42875/42768 ([-4 -5 3 -1 0⟩), 885115/884736 ([-15 -3 1 3 1⟩)

Subgroup-val mapping: [⟨4 5 18 13 18], ⟨0 8 15 5 -6]]

mapping generators: ~2240/1881, ~36/35

Optimal tunings:

  • Subgroup WE: ~2240/1881 = 299.9912 ¢, ~36/35 = 50.2951 ¢
  • Subgroup CWE: ~2240/1881 = 300.0000 ¢, ~36/35 = 50.2947 ¢

Optimal ET sequence: 24, …, 524, 548, 572, 596, 620, 644, 668

Hydrothermal

Named by Budjarn Lambeth in 2024, hydrothermal tempers out 50/49, the jubilisma, in the 2.3.7/5 subgroup. A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, will lend it a mysterious, heavy atmosphere. The 6-tone mos is melodically interesting and flavorful. The 18-tone mos is a useful "chromatic" scale for taking subsets of.

Subgroup: 2.3.7/5

Comma list: 50/49 ([1 0 -2⟩)

Subgroup-val mapping: [⟨2 0 1], ⟨0 1 0]]

mapping generators: ~7/5, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 599.6673 ¢, ~3/2 = 702.5906 ¢
  • Subgroup CWE: ~2 = 600.0000 ¢, ~3/2 = 702.4574 ¢

Optimal ET sequence: 2, 6, 8, 10, 12, 34, 46*

* Wart for 7/5

Argentic

Argentic is the 2.3.7/5-subgroup temperament tempering out 5120/5103, the aberschisma.

Subgroup: 2.3.7/5

Comma list: 5120/5103 ([10 -6 -1⟩)

Subgroup-val mapping: [⟨1 0 10], ⟨0 1 -6]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7172 ¢, ~3/2 = 702.6636 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.8166 ¢

Optimal ET sequence: 12, 29, 41, 70, 321, 391, 461, 531, 601

Badness (Sintel): 0.119

Edson

See also: Chromatic pairs § Edson

Edson is related to pele and andromeda.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 196/195 ([2 -1 2 0 -1⟩), 352/351 ([5 -3 0 1 -1⟩), 364/363 ([2 -1 1 -2 1⟩)

Subgroup-val mapping: [⟨1 0 10 17 22], ⟨0 1 -6 -10 -13]]

Gencom mapping: [⟨1 0 -49/4 -9/4 19/4 39/4], ⟨0 1 29/4 5/4 -11/4 -23/4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1199.4965 ¢, ~3/2 = 703.1192 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 703.4225 ¢

Optimal ET sequence: 12, 17, 29

RMS error: 0.5102 cents

Haumea

See also: Chromatic pairs § Haumea

Haumea is the add-7/5 extension of bridgetown, which is in turn the add-11/5 extension of barbados. Other related temperamnets include hemigari, parizekmic, and namaka.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 352/351 ([5 -3 0 1 -1⟩), 676/675 ([2 -3 0 0 2⟩), 847/845 ([0 0 1 2 -2⟩)

Subgroup-val mapping: [⟨1 0 10 -6 -1], ⟨0 2 -12 9 3]]

Gencom mapping: [⟨1 0 -3/4 37/4 -27/4 -7/4], ⟨0 2 0 -12 9 3]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7072 ¢, ~26/15 = 951.2727 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.5016 ¢

Optimal ET sequence: 24, 29, 111, 140, 169, 198

RMS error: 0.2668 cents

Historical

Not to be confused with Historical temperaments.
Not to be confused with History (temperament).

Historical is essentially an analogue of miracle that splits 4/3 in six rather than 3/2. It tempers out the comma 4000/3993 (S10/S11) to set 11/10 equal to one-third of 4/3, and 676/675 (S13/S15) to equate 15/13 to one-half of 4/3, and tempers out 441/440 (S21) to split 11/10 into two instances of 22/21~21/20. Sextilifourths adds the schismic mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

Subgroup: 2.3.7/5.11/5.13/5

Comma list: 364/363 ([2 -1 1 -2 1⟩), 441/440 ([-3 2 2 -1 0⟩, 1001/1000 ([-3 0 1 1 1⟩)

Subgroup-val mapping: [⟨1 2 0 1 2], ⟨0 -6 7 2 -9]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0242 ¢, ~21/20 = 83.0177 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0144 ¢

Optimal ET sequence: 14, 29, 101, 130, 159

RMS error: 0.2562 cents

Direct breedsmic

This temperament was first proposed by Royalmilktea. The name was established by Lériendil in 2024. Related temperaments: hemififths and newt.

Subgroup: 2.3.49/5

Comma list: 2401/2400 ([-5 -1 2⟩)

Subgroup-val mapping: [⟨1 1 3], ⟨0 2 1]]

mapping generators: ~2, ~49/40

Optimal tuning:

  • Subgroup WE: ~2 = 1200.0206 ¢, ~49/40 = 350.9724 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~49/40 = 350.9743 ¢

Optimal ET sequence: 7, 17, 24, 41, 65, 106, 253, 359, 465, 1036, 1501, 1966*, 5433*

* Wart for 49/5

Tobago

See also: Chromatic pairs § Tobago

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends neutral and barbados.

Subgroup: 2.3.11.13/5

Comma list: 243/242 ([-1 5 -2 0⟩), 676/675 ([2 -3 0 2⟩)

Subgroup-val mapping: [⟨2 0 -1 -2], ⟨0 2 5 3]]

Gencom mapping: [⟨2 0 -5 0 -1 -1], ⟨0 2 -3/2 0 5 3/2]]

mapping generators: ~55/39, ~26/15

Optimal tunings:

  • Subgroup WE: ~55/39 = 599.9845 ¢, ~26/15 = 950.6637 ¢
  • Subgroup CWE: ~55/39 = 600.0000 ¢, ~26/15 = 950.6776 ¢

Optimal ET sequence: 10, 14, 24, 82, 106, 130, 154

RMS error: 0.3533 cents

Pakkanian hemipyth

Subgroup: 2.3.11.13/5.17

Comma list: 221/220 ([-2 0 -1 1 1⟩), 243/242 ([-1 5 -2 0 0⟩), 289/288 ([-5 -2 0 0 2⟩)

Subgroup-val mapping: [⟨2 0 -1 -2 5], ⟨0 2 5 3 2]]

Optimal tunings:

  • subgroup WE: ~17/12 = 600.1781 ¢, ~26/15 = 950.7656 ¢ (~15/13 = 249.2344 ¢)
  • subgroup CWE: ~17/12 = 600.0000 ¢, ~26/15 = 950.6011 ¢ (~15/13 = 249.3989 ¢)

Optimal ET sequence: 10, 14, 24, 106, 130, 154, 178*, 202*

* Wart for 13/5

Bridgetown

See also: Chromatic pairs § Bridgetown

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is the no-7/5 restriction of haumea and the add-11/5 extension of barbados.

Subgroup: 2.3.11/5.13/5

Comma list: 352/351 ([5 -3 1 -1⟩), 676/675 ([2 -3 0 2⟩)

Subgroup-val mapping: [⟨1 0 -6 -1], ⟨0 2 9 3]]

Gencom mapping: [⟨1 0 7/3 0 -11/3 4/3], ⟨0 2 -4 0 5 -1]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • Subgroup WE: ~2 = 1199.6281 ¢, ~26/15 = 951.3060 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.5580 ¢

Optimal ET sequence: 5, 19*, 24, 29, 169, 198, 227, 256, 285, 314, 343

* Wart for 11/5

RMS error: 0.2513 cents

Seventeen-cot

Seventeen-cot may be described as the 29 & 465 temperament in the 2.3.11/5.13/5 subgroup. It tempers out the tendoartisma in the 2.3.13/5 subgroup. It can be generated with a ~2/1 octave and a ~169/165 generator which is 1/17 of a ~3/2 perfect fifth. It was named by FilterNashi in 2026.

Subgroup: 2.3.11/5.13/5

Comma basis: 43940/43923 ([2 -1 -4 3⟩), 225000/224939 ([3 2 -3 -2⟩)

Subgroup-val mapping: [⟨1 1 1 1], ⟨0 17 4 11]]

mapping generators: ~2, ~169/165

Optimal tunings:

  • Subgroup WE: ~2 = 1199.993492 ¢, ~169/165 = 41.291827 ¢
error map: ⟨-0.0065 -0.0004 +0.1566 -0.0104]
  • Subgroup CWE: ~2 = 1200.000000, ~169/165 = 41.291746 ¢
error map: ⟨0.0000 +0.0047 +0.1628 -0.0047]

Optimal ET sequence: 29, 262, 291, 320, 349, 378, 407, 436, 465, 959, 1424, 6161…

Badness (Sintel): 0.080

Blackweed

Blackweed is a restriction of undecimal blackwood as it tempers out 256/243 alike but in the 2.3.11/7 subgroup. 20edo is close to the optimum, which has 4\20 as the period and 420 ¢ as the generator.

Subgroup: 2.3.11/7

Comma list: 256/243 ([8 -5 0⟩)

Subgroup-val mapping: [⟨5 8 0], ⟨0 0 1]]

mapping generators: ~9/8, ~11/7

Optimal tunings:

  • Subgroup WE: ~8/7 = 238.851 ¢, ~11/7 = 782.457 ¢
error map: ⟨-5.746 +8.852 -0.035]
  • Subgroup CWE: ~8/7 = 240.000 ¢, ~11/7 = 784.967 ¢
error map: ⟨0.000 +18.045 +2.475]

Optimal ET sequence: 15, 20, 35b, 55b

Pepperoni

Main article: Parapyth
See also: Chromatic pairs § Pepperoni

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of parapyth. The Pepper fifth, which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

Subgroup: 2.3.11/7.13/7

Comma list: 352/351 ([5 -3 1 -1⟩), 364/363 ([2 -1 -2 1⟩)

Subgroup-val mapping: [⟨1 0 7 12], ⟨0 1 -4 -7]]

Gencom mapping: [⟨1 0 0 -19/3 2/3 17/3], ⟨0 1 0 11/3 -1/3 -10/3]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.3706 ¢, ~11/7 = 703.4872 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~11/7 = 703.8328 ¢

Optimal ET sequence: 12, 17, 29, 75, 104

RMS error: 0.3789 cents

Fendo

Fendo tempers out 40/39 in the 2.3.13/5 subgroup. This equates 4/3 with the tendo third 13/10. Fendo can be viewed as an alternative interpretation of scales, chords, etc. commonly associated with father, as it is sometimes perceived that father's characteristics tend to suggest harmonic function more than to actually serve as a temperament. It was named by Vector in 2025 as a contraction of fourth and tendo third.

In fendo, the fourth / tendo third serves the role of both, making the main chords of triadic harmony in fendo essentially suspended chords.

5edo and 13edo make good fendo tunings.

Subgroup: 2.3.13/5

Comma list: 40/39 ([3 -1 -1⟩)

Subgroup-val mapping: [⟨1 0 3], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1195.6502 ¢, ~3/2 = 709.8204 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 710.1529 ¢

Optimal ET sequence: 1, 2, 3, 5, 17*, 22*, 27*, 32*

* Wart for 13/5

Barbados

Barbados is the restriction of bridgetown to the 2.3.13/5 subgroup, and tempers out 676/675 alone. The minimax tuning for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are 24edo, 29edo, 53edo and 111edo, with mos scales of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

Subgroup: 2.3.13/5

Comma list: 676/675 ([2 -3 2⟩)

Subgroup-val mapping: [⟨1 0 -1], ⟨0 2 3]]

mapping generators: ~2, ~26/15

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9502 ¢, ~26/15 = 951.0713 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.0933 ¢

Optimal ET sequence: 5, 14, 19, 24, 29, 53, 82, 135

Badness (Smith): 0.002335

Music

Fiventeen

Fiventeen tempers out 136/135 ([3 -3 1⟩) in 2.3.17/5. It equates 17/15 with 9/8, so it implies a supersoft pentic pentad of ~30:34:40:45:51. 17edo makes a good tuning especially for its size, which gives a supersoft pentic scale corresponding approximately to a just 20/17 tuning, although 80edo might be preferred for an approximately just 51/40 to optimize plausibility slightly more, and 97edo (= 80 + 17) and 114edo (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the optimal ET sequence is much more characteristic of optimized tunings, finding 34edo, then 80edo, then 114edo (= 34 + 80) and even 194bc-edo (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting 63edo and 143edo (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

Subgroup: 2.3.17/5

Comma list: 136/135 ([3 -3 1⟩)

Subgroup-val mapping: [⟨1 0 -3], ⟨0 1 3]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.2838 ¢, ~3/2 = 704.4600 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 704.5286 ¢

Optimal ET sequence: 5, 12, 17, 46, 63, 143

Surprise

This temperament was named by Vector in 2025, as he was surprised that the temperament of 57/56 did not have a name. This is the rank-2 version of the temperament; Vector surmises that the name hendrix would be more thoughtfully given to the rank-3 version.

Subgroup: 2.3.19/7

Comma list: 57/56 ([-3 1 1⟩)

Subgroup-val mapping: [⟨1 0 3], ⟨0 1 -1]]

mapping generators: ~2, ~3

Optimal tunings:

  • Subgroup WE: ~2 = 1202.4345 ¢, ~3/2 = 697.4314 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~3/2 = 697.3981 ¢

Optimal ET sequence: 5, 7, 12, 19, 31*, 50*

* Wart for 19/7

Badness (Sintel): 0.082

2.9.… subgroups

See also antikythera and isra.

Baldy

See also: Chromatic pairs § Baldy

Baldy is every other step of garibaldi. It is also the add-9 extension of frostburn. One of the best extension is to the 2.9.5.7.13 subgroup, mapping 13/8 to +10 whole tones, the same as the cassandra temperament.

Subgroup: 2.9.5.7

Comma list: 225/224, 3125/3087

Subgroup-val mapping: [⟨1 0 15 25], ⟨0 1 -4 -7]]

mapping generators: ~2, ~9

Optimal tunings:

  • WE: ~2 = 1200.1233 ¢, ~9/8 = 204.1913 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.1549 ¢

Optimal ET sequence: 6, 29, 35, 41, 47, 194

Badness (Sintel): 0.274

2.9.5.7.13 subgroup

Subgroup: 2.9.5.7.13

Comma list: 225/224, 325/324, 640/637

Subgroup-val mapping: [⟨1 0 15 25 -28], ⟨0 1 -4 -7 10]]

Gencom mapping: [⟨1 0 15 25 0 28], ⟨0 1/2 -4 -7 0 10]]

Optimal tunings:

  • WE: ~2 = 1200.0155 ¢, ~9/8 = 204.0931 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.0902 ¢

Optimal ET sequence: 6, 35, 41, 47, 100, 147

Badness (Sintel): 0.386

RMS error: 0.5999 cents

Baldanders

Baldanders results from taking every other generator of the andromeda, mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

Subgroup-val mapping: [⟨1 0 15 25 32], ⟨0 1 -4 -7 -9]]

Optimal tunings:

  • WE: ~2 = 1200.1917 ¢, ~9/8 = 204.7755 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.7197 ¢

Optimal ET sequence: 6, 23de, 29, 35, 41

Badness (Sintel): 0.389

2.9.5.7.11.13

Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

Subgroup-val mapping: [⟨1 0 15 25 32 -28], ⟨0 1 -4 -7 -9 10]]

Optimal tunings:

  • WE: ~2 = 1199.6385 ¢, ~9/8 = 204.3526 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 204.4234 ¢

Optimal ET sequence: 6, 35, 41, 47, 88e

Badness (Sintel): 0.653

Daemotertiaschis

See also: Schismatic family § Tertiaschis

Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a daemotonic 7L 4s scale of reasonable hardness (hence the name – daemo- + tertiaschis), which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

Subgroup-val mapping: [⟨1 1 11 -16 13 -18 20], ⟨0 3 -12 26 -11 30 -22]]

mapping generators: ~2, ~33/20

Optimal tunings:

  • Subgroup WE: ~2 = 1200.3019 ¢, 33/20 = 868.1949 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, 33/20 = 867.9773 ¢

Optimal ET sequence: 47, 159, 206, 253, 459…

Badness (Sintel): 0.439

A-team

See also: Dual-fifth temperaments § Dual-3 A-Team

A-team is every other step of mothra.

Subgroup: 2.9.5.21

Comma list: 81/80 ([-4 2 -1 0⟩), 1029/1024 ([-10 -1 0 3⟩)

Subgroup-val mapping: [⟨1 2 0 4], ⟨0 3 6 1]]

Gencom mapping: [⟨1 1 0 3], ⟨0 3/2 6 -1/2]]

mapping generators: ~2, ~21/16

Optimal tunings:

  • Subgroup WE: ~2 = 1200.9303 ¢, ~21/16 = 464.7465 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 464.5029 ¢

Optimal ET sequence: 13, 18, 31

Badness (Sintel): 0.177

2.9.5.21.11 subgroup

Subgroup: 2.9.5.21.11

Comma list: 81/80 ([-4 2 -1 0⟩), 99/98 ([-1 2 0 -2 1⟩), 385/384 ([-7 -1 1 1 1⟩)

Subgroup-val mapping: [⟨1 2 0 4 5], ⟨0 3 6 1 -4]]

Gencom mapping: [⟨1 1 0 3 5], ⟨0 3/2 6 -1/2 -4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1201.3979 ¢, ~21/16 = 464.6020 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 464.1242 ¢

Optimal ET sequence: 13, 31, 44, 75b′

Badness (Sintel): 0.213

Glacial

See also: Chromatic pairs § Glacial

Subgroup: 2.9.5.11.13

Comma list: 45/44, 65/64, 81/80

Subgroup-val mapping: [⟨1 0 -4 -6 10], ⟨0 1 2 3 -2]]

Gencom mapping: [⟨1 0 -4 0 -6 10], ⟨0 1/2 2 0 3 -2]]

mapping generators: ~2, ~9

Optimal tunings:

  • WE: ~2 = 1202.6916 ¢, ~9/8 = 186.1446 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 186.1740 ¢

Optimal ET sequence: 6, 13, 45b′f

Badness (Sintel): 0.281

RMS error: 2.887 cents

Music:

Stacks (a.k.a. 2magic)

Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of magic.

Subgroup: 2.9.15.7

Comma list: 225/224 ([-5 0 2 -1⟩), 245/243 ([0 -3 1 2⟩)

Subgroup-val mapping: [⟨1 0 2 -1], ⟨0 5 3 6]]

Gencom mapping: [⟨1 0 2 -1], ⟨0 5/2 1/2 6]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • Subgroup WE: ~2 = 1201.0786 ¢, ~14/9 = 761.3878 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~14/9 = 760.9151 ¢

Optimal ET sequence: 8, 11, 19, 30, 41, 71, 112*

* Wart for 15

Badness (Sintel): 0.0884

RMS error: 1.074 cents

2.9.15.7.11

Subgroup: 2.9.15.7.11

Comma list: 100/99 ([2 -2 2 0 -1⟩), 225/224 ([-5 0 2 -1 0⟩), 245/243 ([0 -3 1 2 0⟩)

Subgroup-val mapping: [⟨1 0 2 -1 6], ⟨0 5 3 6 -4]]

Gencom mapping: [⟨1 0 2 -1 6], ⟨0 5/2 1/2 6 -4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.1372 ¢, ~14/9 = 761.4799 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~14/9 = 761.4017 ¢

Optimal ET sequence: 8, 11, 30, 41, 52, 93, 145

Badness (Sintel): 0.166

RMS error: 1.226 cents

2.9.15.7.11.13

Subgroup: 2.9.15.7.11.13

Comma list: 100/99 ([2 -2 2 0 -1 0⟩), 105/104 ([-3 0 1 1 0 -1⟩), 144/143 ([4 1 0 0 -1 -1⟩), 196/195 ([2 0 -1 2 0 -1⟩)

Subgroup-val mapping: [⟨1 0 2 -1 6 -2], ⟨0 5 3 6 -4 9]]

Gencom mapping: [⟨1 0 2 -1 6 -2], ⟨0 5/2 1/2 6 -4 9]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0331 ¢, ~14/9 = 760.8754 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~14/9 = 760.8568 ¢

Optimal ET sequence: 11, 30, 41

Badness (Sintel): 0.257

RMS error: 1.540 cents

B-team

B-team may be described as 23 & 41 and is every other step of rodan.

Subgroup: 2.9.15.21

Comma list: 245/243 ([0 -4 1 2⟩), 1029/1024 ([-10 -1 0 3⟩)

Subgroup-val mapping: [⟨1 2 0 4], ⟨0 3 10 1]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.2146 ¢, ~21/16 = 468.9174 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 468.8518 ¢

Optimal ET sequence: 18, 23, 41, 64, 151

Badness (Sintel): 0.155

2.9.15.21.33 subgroup

Subgroup: 2.9.15.21.33

Comma list: 245/243 ([0 -4 1 2 0⟩), 385/384 ([-7 -2 1 1 1⟩), 441/440 ([-3 1 -1 2 -1⟩)

Subgroup-val mapping: [⟨1 2 0 4 7], ⟨0 3 10 1 -5]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0553 ¢, ~21/16 = 468.9391 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/16 = 468.9189 ¢

Optimal ET sequence: 18, 23, 41, 64, 87, 151, 238b′

Badness (Sintel): 0.160

Mabon

Mabon may be described as the 11 & 62 temperament in the 2.9.7 subgroup. It was derived from a calendar leap cycle built for the autumn equinox, and named so by Dummy index in 2024.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Subgroup-val mapping: [⟨1 1 -3], ⟨0 3 8]]

mapping generators: ~2, ~729/448

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~729/448 = 870.4003 ¢
  • CWE: ~2 = 1200.0000 ¢, ~729/448 = 871.1047 ¢

Optimal ET sequence: 7d, 11, 40, 51, 62, 73b′, 157b′b′

Badness (Sintel): 4.08

2.9.7.11 subgroup

Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Subgroup-val mapping: [⟨1 1 -3 2], ⟨0 3 8 2]]

Optimal tunings:

  • WE: ~2 = 1200.1182 ¢, ~18/11 = 870.9904 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 870.9234 ¢

Optimal ET sequence: 7d, 11, 40, 51, 62

Badness (Sintel): 4.08

Machine

Machine is every other step of supra, most interesting for its scale patterns.

It was named by Mike Battaglia in 2011, who spoke highly of this temperament, describing it as "stimulating, like a well-oiled machine".[1]

Subgroup: 2.9.7.11

Comma list: 64/63, 99/98

Subgroup-val mapping: [⟨1 0 6 13], ⟨0 1 -1 -3]]

Gencom mapping: [⟨1 0 0 6 13], ⟨0 1/2 0 -1 -3]]

mapping generators: ~2, ~9

Optimal tunings:

  • WE: ~2 = 1197.2650 ¢, ~9/8 = 213.8957 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/8 = 214.9961 ¢

Optimal ET sequence: 5, 6, 11, 17, 28

Badness (Sintel): 0.176

Apparatus

Apparatus is the restriction of revelation to the 2.9.7.11 subgroup, and every fourth step of quasiorwell, without prime 5.

Subgroup: 2.9.7.11

Comma list: 41503/41472, 322102/321489

Subgroup-val mapping: [⟨1 -14 1 -11], ⟨0 19 2 16]]

Gencom mapping: [⟨1 -7 0 1 -11], ⟨0 19/2 0 2 16]]

mapping generators: ~2, ~144/77

Optimal tunings:

  • WE: ~2 = 1200.0088 ¢, ~144/77 = 1084.4377 ¢
  • CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.4304 ¢

Optimal ET sequence: 10e, 21, 31, 52, 83, 135, 353, 488, 623

Badness (Sintel): 0.199

Penta a.k.a. mechanism

Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

Subgroup: 2.9.7.11

Comma list: 896/891, 26411/26244

Subgroup-val mapping: [⟨1 0 -1 6], ⟨0 5 6 -4]]

Gencom mapping: [⟨1 0 0 -1 6], ⟨0 5/2 0 6 -4]]

mapping generators: ~2, ~14/9

Optimal tunings:

  • WE: ~2 = 1199.4843 ¢, ~14/9 = 761.2079 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/9 = 761.4979 ¢

Optimal ET sequence: 8, 11, 30, 41, 52

Badness (Sintel): 0.331

RMS error: 0.4262 cents

Scales: penta5, penta8, penta11, penta19

Joan

See also: Chromatic pairs § Joan

Joan is the restriction of casablanca to the 2.9.7.11 subgroup, as well as every other step of orwell, without prime 5.

Subgroup: 2.9.7.11

Comma list: 99/98, 9317/9216

Subgroup-val mapping: [⟨1 0 1 3], ⟨0 7 4 1]]

Gencom mapping: [⟨1 0 0 1 3], ⟨0 7/2 0 4 1]]

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1200.8640 ¢, ~11/8 = 543.0623 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 542.8446 ¢

Optimal ET sequence: 11, 20, 31, 42, 115b′d

Badness (Sintel): 0.407

RMS error: 1.424 cents

Novisept

Named by Pailiaq in 2026, novisept is generated by a one-cent-flat 9/7, such that stacking 5 of them gives 7/4. It can be formed by doubling both generator and period of gizzard.

Subgroup: 2.9.7.13.17

Comma list: 442/441, 729/728, 833/832

Subgroup-val mapping: [⟨1 1 1 -1 3], ⟨0 6 5 13 3]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • WE: ~2 = 1200.3044 ¢, ~9/7 = 433.9023 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 433.8360 ¢

Optimal ET sequence: 11, 25f, 36, 47, 83, 130, 213g, 296g

Badness (Sintel): 0.142

Genius

Genius tempers out the Alpharabian comma in the 2.9.11 subgroup; it is therefore every other step of alphaxenean. It was named by Akselai in 2024 after the genius in Roman religion, following the demon (daimon) in Greek mythology.

Subgroup: 2.9.11

Comma list: 131769/131072

Subgroup-val mapping: [⟨1 1 4], ⟨0 4 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • WE: ~2 = 1200.3181 ¢, ~16/11 = 650.8508 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 650.7321 ¢

Optimal ET sequence: 11, 24, 59, 83, 142, 225, 367e, 592b′e, 959b′eee, 1326b′b′eeee

Badness (Sintel): 0.198

Demon

Demon is a temperament which equates three 11/9's with 16/9, or equivalently three 18/11's with 9/8, tempering out 1331/1296. This results in 11/9 being tuned flat to a supraminor third, and 27/22 being tuned sharp to a submajor third. It was discovered by CompactStar while searching for temperaments assosciated with the 7L 4s ("daemotonic") mos, known for its lack of representation of simple temperaments. The optimal tuning for the demon temperament is near the basic tuning of 7L 4s (13\18), and indeed 18edo supports this temperament.

Mabon is a strong extension of this temperament to the 2.9.7.11 subgroup.

Subgroup: 2.9.11

Comma list: 1331/1296

Subgroup-val mapping: [⟨1 1 2], ⟨0 3 2]]

mapping generators: ~2, ~18/11

Optimal tunings:

  • WE: ~2 = 1201.1151 ¢, ~18/11 = 869.4724 ¢
  • CWE: ~2 = 1200.0000 ¢, ~18/11 = 869.3196 ¢

Optimal ET sequence: 4, 7, 11, 18, 29, 76e

Badness (Sintel): 0.332

4.3.… subgroups

Tetrameantone

Main article: Tetrameantone

Named by Budjarn Lambeth in 2024, tetrameantone is the double-octave restriction of meantone.

Subgroup: 4.3.5

Comma list: 81/80

Subgroup-val mapping: [⟨1 0 -2], ⟨0 1 4]]

mapping generators: ~4, ~3

Optimal tunings:

  • WE: ~4 = 2402.7812 ¢, ~3 = 1898.4361 ¢
  • CWE: ~4 = 2400.0000 ¢, ~3 = 1896.6512 ¢

Optimal ET sequence: a′5, a′14, a′19, a′43, a′62, a′81, a′262b, a′343b

Badness (Sintel): 0.0866

Tetrahanson

Main article: Tetrahanson

Named by CompactStar in 2023, tetrahanson is the double-octave restriction of hanson.

Subgroup: 4.3.5

Comma list: 15625/15552

Subgroup-val mapping: [⟨1 -3 -2], ⟨0 6 5]]

mapping generators: ~4, ~12/5

Optimal tunings:

  • WE: ~4 = 2400.3318 ¢, ~12/5 = 1517.2163 ¢
  • CWE: ~4 = 2400.0000 ¢, ~12/5 = 1517.0308 ¢

Optimal ET sequence: a′8, a′11, a′19, a′68, a′87, a′106, a′511, a′617, a′723, a′829, a′935c, a′1764c

Badness (Sintel): 0.155

Tetramagic

Named by Budjarn Lambeth in 2024, tetramagic is the double-octave restriction of magic.

Subgroup: 4.3.5

Comma list: 3125/3072

Subgroup-val mapping: [⟨1 0 1], ⟨0 5 1]]

mapping generators: ~4, ~5/4

Optimal tunings:

  • WE: ~4 = 2402.4898 ¢, ~5/4 = 380.4527 ¢
  • CWE: ~4 = 2400.0000 ¢, ~5/4 = 380.2194 ¢

Optimal ET sequence: a′6, a′13, a′19, a′63, a′82, a′101, a′120, a′221c, a′341c

Badness (Sintel): 0.459

Blacktetra

Named by Budjarn Lambeth in 2024, blacktetra is the double-octave restriction of blackwood.

Subgroup: 4.3.5

Comma list: 256/243

Subgroup-val mapping: [⟨5 4 0], ⟨0 0 1]]

mapping generators: ~4, ~5

Optimal tunings:

  • WE: ~4/3 = 477.7018 ¢, ~5/4 = 397.6806 ¢ (~16/15 = 80.0212 ¢)
  • CWE: ~4/3 = 480.0000 ¢, ~5/4 = 395.1255 ¢ (~16/15 = 84.7745 ¢)

Optimal ET sequence: a′5, a′20, a′25, a′30

Badness (Sintel): 0.748

4.6.… subgroups

Meanquad

Meanquad is the double-octave restriction of meantone to the 4.6.5 subgroup. Similarly to meantone, it has a flat 3/2 as generator and equates 5/4 and 81/64. It possesses mos scales of the forms 3L 4s<4/1>, 7L 3s<4/1>, and 7L 10s<4/1>, which could be viewed as equivalents to the pentic, diatonic, and chromatic scales of meantone, respectively.

17ed4 is one of the smallest tunings that support this temperament and features the decatonic scale. Like 12edo, it can be viewed as the boundary between meanquad and quarchy, the 4.6.7-subgroup restriction of archy. Larger options include 24ed4 (12edo), 31ed4 and 55ed4.

Subgroup: 4.6.5

Comma list: 81/80 ([-4 4 -1⟩)

Subgroup-val mapping: [⟨1 0 -4], ⟨0 1 4]]

mapping generators: ~4, ~6

Optimal tunings:

  • Subgroup WE: ~4 = 2402.7812 ¢, ~3/2 = 697.0455 ¢
  • Subgroup CWE: ~4 = 2400.0000 ¢, ~3/2 = 696.6512 ¢

Optimal ET sequence: a′7, a′17, a′24, a′31, a′100, a′131, a′293*

* Wart for 6

Badness (Sintel): 0.0146

Tetrominant

Subgroup: 4.6.5.7

Comma list: 36/35 ([0 2 -1 -1⟩), 64/63 ([4 -2 0 -1⟩)

Subgroup-val mapping: [⟨1 0 -4 4], ⟨0 1 4 -2]]

Optimal tunings:

  • Subgroup WE: ~4 = 2400.0000 ¢, ~3/2 = 699.622 ¢
  • Subgroup CWE: ~4 = 2400.0000 ¢, ~3/2 = 699.622 ¢

Optimal ET sequence: a′7, a′17, a′24, a′41, a′65cd

Badness (Sintel): 0.0974

Quarchy

Subgroup: 4.6.7

Comma list: 64/63 ([4 -2 -1⟩)

Subgroup-val mapping: [⟨1 0 4], ⟨0 1 -2]]

mapping generators: ~4, ~6

Optimal tunings:

  • Subgroup WE: ~4 = 2393.9105 ¢, ~3/2 = 707.5215 ¢
  • Subgroup CWE: ~4 = 2400.0000 ¢, ~3/2 = 709.3901 ¢

Optimal ET sequence: a′7, a′10, a′17, a′27, a′44, a′159d, a′203*dd

* Wart for 6

Badness (Sintel): 0.0134

4.9.… subgroups

Meansquared

Named by CompactStar in 2023, meansquared tempers out the double syntonic comma, 6561/6400, in the 4.9.25 subgroup.

Subgroup: 4.9.25

Comma list: 6561/6400

Subgroup-val mapping: [⟨1 0 -4], ⟨0 1 4]]

mapping generators: ~4, ~9

Optimal tunings:

  • WE: ~4 = 2402.7812 ¢, ~9/4 = 1394.0910 ¢
  • CWE: ~4 = 2400.0000 ¢, ~9/4 = 1393.3025 ¢

Optimal ET sequence: a′5, a′7, a′12, a′19, a′31, a′50, a′81, a′131b′

Badness (Sintel): 0.173

Archsquared

Named by CompactStar in 2023, archsquared tempers out the double archytas comma, 4096/3969, in the 4.9.49 subgroup.

Subgroup: 4.9.49

Comma list: 4096/3969

Subgroup-val mapping: [⟨1 0 6], ⟨0 1 -2]]

mapping generators: ~4, ~9

Optimal tunings:

  • WE: ~4 = 2393.9105 ¢, ~9/4 = 1415.0430 ¢
  • CWE: ~4 = 2400.0000 ¢, ~9/4 = 1418.7801 ¢

Optimal ET sequence: a′5, a′12, a′17, a′22, a′137b′d′d′, a′159b′d′d′d′, a′181b′b′d′d′d′

Badness (Sintel): 0.159

8.9.… subgroups

Sixscared

Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, sixscared was named by Budjarn Lambeth in 2023 for the classic dad joke: "Why was six scared? Because seven ate nine."

Subgroup: 8.9.7

Comma list: 64/63

Subgroup-val mapping: [⟨1 0 2], ⟨0 1 -1]]

mapping generators: ~8, ~9

Optimal tunings:

  • WE: ~8 = 3590.8657 ¢, ~9/8 = 219.0878 ¢
  • CWE: ~8 = 3600.0000 ¢, ~9/8 = 218.7801 ¢

Optimal ET sequence: a″16, a″33, a″148b′d, a″181b′d, a″214b′d, a″247b′b′d

Badness (Sintel): 0.0265

2.5.… subgroups

Guanyintet

See also: Chromatic pairs § Guanyintet

Guanyintet, the 4 & 9 temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of guanyin and a restriction of orwell. It is defined by tempering out 176/175 and 540/539, which imply 1728/1715 as well as 3388/3375 are tempered out. The tonic and the first three generator steps make a guanyin tetrad, hence the name.

Subgroup: 2.5.7/3.11/3

Comma list: 176/175 ([4 -2 -1 1⟩), 540/539 ([2 1 -2 -1⟩)

Subgroup-val mapping: [⟨1 0 1 3], ⟨0 3 -1 5]]

Gencom mapping: [⟨1 0 0 2 -2], ⟨0 -4/3 3 -7/3 11/3]]

mapping generators: ~2, ~12/7

Optimal tunings:

  • Subgroup WE: ~2 = 1199.2336 ¢, ~12/7 = 929.3128 ¢ (~7/6 = 269.9208 ¢)
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/7 = 929.7608 ¢ (~7/6 = 270.2392 ¢)

Optimal ET sequence: 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c*

* Wart for 7/3

RMS error: 0.6028 cents

Tridecimal guanyintet

Guanyintet can extend to the add-13 subgroup by the equivalences (12/11)3~13/10 and (15/14)3~16/13. However, note that it is not supported by tridecimal orwell (31 & 53), but instead the less accurate winston (22f & 31), as orwell prefers slightly sharper tunings than guanyintet. 40edo remains an excellent tuning.

Subgroup: 2.5.7/3.11/3.13

Comma list: 176/175 ([4 -2 -1 1 0⟩), 540/539 ([2 1 -2 -1 0⟩), 1573/1568 ([-5 0 -2 2 1⟩)

Subgroup-val mapping: [⟨1 0 1 3 1], ⟨0 3 -1 5 -12]]

Optimal tunings:

  • Subgroup WE: ~2 = 1199.3604 ¢, ~12/7 = 929.2866 ¢ (~7/6 = 270.0738 ¢)
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/7 = 929.7980 ¢ (~7/6 = 270.2020 ¢)

Optimal ET sequence: 9, 22f, 31, 40, 71, 111, 151, 262c*

* Wart for 7/3

Badness (Sintel): 0.329

Laz

See also: Chromatic pairs § Laz

Laz is related to avalokita as well as to winston.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 144/143 ([4 0 0 -1 -1⟩), 176/175 ([4 -2 -1 1⟩), 196/195 ([2 -1 2 0 -1⟩

Subgroup-val mapping: [⟨1 0 2 -2 6], ⟨0 3 -1 5 -5]]

Gencom mapping: [⟨1 -3/2 0 1/2 -7/2 9/2], ⟨0 1/4 3 -3/4 21/4 -19/4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1198.4320 ¢, ~12/7 = 929.4834 ¢ (~7/6 = 268.9486 ¢)
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/7 = 930.5797 ¢ (~7/6 = 269.4203 ¢)

Optimal ET sequence: 9, 31, 40, 49, 156c*†, 205c*†

* Wart for 7/3
† Wart for 11/3

RMS error: 0.8790 cents

Kryptonite

See also: Chromatic pairs § Kryptonite

Kryptonite is related to krypton.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 56/55 ([3 -1 1 -1⟩), 78/77 ([1 0 -1 -1 1⟩), 91/90 ([-1 -2 1 0 1⟩)

Subgroup-val mapping: [⟨1 2 1 2 2], ⟨0 3 2 -1 1]]

Gencom mapping: [⟨1 -5/4 2 -1/4 3/4 3/4], ⟨0 -1/2 3 3/2 -3/2 1/2]]

mapping generators: ~2, ~13/12

Optimal tunings:

  • Subgroup WE: ~2 = 1197.6484 ¢, ~13/12 = 132.1686 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/12 = 131.8445 ¢

Optimal ET sequence: 1, …, 8, 9

RMS error: 2.545 cents

Marveltri

See also: Chromatic pairs § Marveltri

Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament. The tonic and the first two generator steps make a marvel triad, hence the name.

Subgroup: 2.5.9/7

Comma list: 225/224 ([-5 2 1⟩)

Subgroup-val mapping: [⟨1 0 5], ⟨0 1 -2]]

Gencom mapping: [⟨1 2 0 -1], ⟨0 -4/5 1 2/5]]

mapping generators: ~2, ~5

Optimal tunings:

  • Subgroup WE: ~2 = 1200.5971 ¢, ~5/4 = 383.8285 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/4 = 383.8540 ¢

Optimal ET sequence: 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c*

* Wart for 9/7

RMS error: 0.4801 cents

Sulis

Sulis is related to minerva and würschmidt.

Subgroup: 2.5.9/7.11/9

Comma list: 99/98 ([-1 0 2 1⟩), 176/175 ([4 -2 1 1⟩)

Subgroup-val mapping: [⟨1 0 5 -9], ⟨0 1 -2 4]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.1086 ¢, ~5/4 = 386.5931 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/4 = 386.5785 ¢

Optimal ET sequence: 3, …, 22, 25, 28, 31, 59

RMS error: 1.074 cents

2.5/3.… subgroups

Magicaltet

See also: Chromatic pairs § Magicaltet

Magicaltet is related to keemic, superkleismic, and magic. The tonic and the first three generator steps make a magical seventh chord, hence the name.

Subgroup: 2.5/3.7.11

Comma list: 100/99 ([2 2 0 -1⟩), 385/384 ([-7 1 1 1⟩)

Subgroup-val mapping: [⟨1 0 5 2], ⟨0 1 -3 2]]

Gencom mapping: [⟨1 0 0 5 2], ⟨0 -1/2 1/2 -3 2]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9868 ¢, ~5/3 = 877.3312 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 877.3417 ¢

Optimal ET sequence: 4, 7, 11, 15, 26, 67, 93*

* Wart for 5/3

RMS error: 1.206 cents

Starlingtet

See also: Chromatic pairs § Starlingtet

Starlingtet, the 4 & 15 temperament in the 2.5/3.7/3 subgroup, is related to starling as well as to myna. The tonic and the first three generator steps make a starling tetrad, hence the name.

Subgroup: 2.5/3.7/3

Comma list: 126/125 ([1 -3 1⟩)

Subgroup-val mapping: [⟨1 0 -1], ⟨0 1 3]]

Gencom mapping: [⟨1 1/3 1/3 -2/3], ⟨0 -4/3 -1/3 5/3]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7931 ¢, ~5/3 = 888.6923 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 888.8022 ¢

Optimal ET sequence: 4, 15, 19, 23, 27

RMS error: 0.8398 cents

Skateboard

See also: Chromatic pairs § Skateboard

Skateboard is related to thrasher.

Subgroup: 2.5/3.7/3.11.13/9

Comma list: 56/55 ([3 -1 1 -1⟩), 91/90 ([-1 -1 1 0 1⟩), 100/99 ([2 2 0 -1⟩)

Subgroup-val mapping: [⟨1 0 -1 2 2], ⟨0 1 3 2 -2]]

Gencom mapping: [⟨1 -3/7 -3/7 -10/7 2 8/7], ⟨0 0 1 3 2 -2]]

Optimal tunings:

  • Subgroup WE: ~2 = 1197.4723 ¢, ~5/3 = 884.2917 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 885.6469 ¢

Optimal ET sequence: 4, 11, 15, 19, 23, 42e

RMS error: 2.396 cents

Greeley

See also: Chromatic pairs § Greeley

Greeley is related to opossum as well as to nusecond.

Subgroup: 2.5/3.7/3.11/3

Comma list: 121/120 ([-3 -1 0 2⟩), 126/125 ([1 -3 1⟩)

Subgroup-val mapping: [⟨1 1 2 2], ⟨0 -2 -6 -1]]

Gencom mapping: [⟨1 -5/4 -1/4 3/4 3/4], ⟨0 9/4 1/4 -15/4 5/4]]

mapping generators: ~2, ~11/10

Optimal tunings:

  • Subgroup WE: ~2 = 1200.4124 ¢, ~11/10 = 155.8292 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~11/10 = 155.7399 ¢

Optimal ET sequence: 8, 15, 23, 54, 77, 131*

* Wart for 11/3

RMS error: 1.034 cents

Gariberttet

Gariberttet is the 2.5/3.7/3 altergene[idiosyncratic term][clarification needed] of sirius, tempering out 3125/3087. A notable tuning besides WE and CWE given below is CTE/POTE at 293.6785 cents.

Subgroup: 2.5/3.7/3

Comma list: 3125/3087

Subgroup-val mapping: [⟨1 0 0], ⟨0 3 5]]

mapping generators: ~2, ~25/21

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9983 ¢, ~25/21 = 293.7393 ¢
error map: ⟨-0.002 -3.141 1.826]
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~25/21 = 293.7395 ¢
error map: ⟨0.000 -3.140 1.827]

Optimal ET sequence: 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143, 192

* Wart for 7/3

2.5/3.7/3.13/11 subgroup

See also: Chromatic pairs § Gariberttet

Gariberttet can be described as the 4 & 29 temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include quasitemp.

Subgroup: 2.5/3.7/3.13/11

Comma list: 275/273 ([0 2 -1 -1⟩), 847/845 ([0 -1 1 -2⟩)

Subgroup-val mapping: [⟨1 0 0 0], ⟨0 3 5 1]]

Gencom mapping: [⟨1 0 0 0 0 0], ⟨0 -8/3 1/3 7/3 -1/2 1/2]]

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9976 ¢, ~13/11 = 293.6780 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/11 = 293.6782 ¢

Optimal ET sequence: 4, 25*, 29, 33, 37, 41, 45, 49, 94, 143†

* Wart for 7/3
† Wart for 13/11

RMS error: 0.6914 cents

Indium

See also: Chromatic pairs § Indium

Indium can be described as the 8 & 33 temperament in the 2.5/3.7/3.11/3 subgroup.

Subgroup: 2.5/3.7/3.11/3

Comma list: 3025/3024 ([-4 2 -1 2⟩), 3125/3087 ([0 5 -3⟩)

Subgroup-val mapping: [⟨1 0 0 2], ⟨0 6 10 -1]]

Gencom mapping: [⟨1 -1/2 -1/2 -1/2 3/2], ⟨0 -15/4 9/4 25/4 -19/4]]

mapping generators: ~2, ~12/11

Optimal tunings:

  • Subgroup WE: ~2 = 1199.5499 ¢, ~12/11 = 146.9551 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~12/11 = 146.9960 ¢

Optimal ET sequence: 8, 33, 41, 49, 204*†

* Wart for 7/3
† Wart for 11/3

RMS error: 0.7788 cents

Ammon

See also: Chromatic pairs § Ammon

Ammon can be described as the 8 & 29 temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends tridec, and is related to ammonite. It is generated by ~20/13, whose octave complement, ~13/10, may be considered a semidiminished fourth, hence the old name semidim, which has been rejected since 2025 to avoid confusion with another temperament of the same name.

Subgroup: 2.5/3.7/3.11/3.13/3

Comma list: 121/120 ([-3 -1 0 2⟩), 169/168 ([-3 0 -1 0 2⟩), 275/273 ([0 2 -1 1 -1⟩)

Subgroup-val mapping: [⟨1 -3 -5 0 -1], ⟨0 6 10 3 5]]

Gencom mapping: [⟨1 9/5 -6/5 -16/5 9/5 4/5], ⟨0 -24/5 6/5 26/5 -9/5 1/5]]

mapping generators: ~2, ~20/13

Optimal tunings:

  • Subgroup WE: ~2 = 1200.9125 ¢, ~20/13 = 747.3256 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~20/13 = 746.8104 ¢

Optimal ET sequence: 8, 29, 37, 45

RMS error: 1.052 cents

Sentry

See also: Chromatic pairs § Sentry

Sentry, the 3 & 5 temperament in the 2.5/3.9/7 subgroup, tempers out 245/243 and shares the same generator with sensi. A notable tuning besides WE and CWE given below is CTE/POTE at 440.9020 cents.

Subgroup: 2.5/3.9/7

Comma list: 245/243 ([0 1 -2⟩)

Subgroup-val mapping: [⟨1 0 0], ⟨0 2 1]]

Gencom mapping: [⟨1 0 0 0], ⟨0 0 2 -1]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9983 ¢, ~9/7 = 440.9014 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~9/7 = 440.9017 ¢

Optimal ET sequence: 3, 5, 8, 11, 19, 30, 41, 49

RMS error: 0.7105 cents

Marveltwintri

See also: Chromatic pairs § Marveltwintri

Marveltwintri can be described as the 3 & 4 temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a marveltwin triad, hence the name. Cata is a very natural extension of this temperament to the 2.3.5.13 subgroup.

Subgroup: 2.5/3.13/9

Comma list: 325/324 ([-2 2 1⟩)

Subgroup-val mapping: [⟨1 0 2], ⟨0 1 -2]]

Gencom mapping: [⟨1 -2/3 -7/3 0 0 2/3], ⟨0 1/2 3/2 0 0 -1]]

mapping generators: ~2, ~5/3

Optimal tunings:

  • Subgroup WE: ~2 = 1200.1340 ¢, ~5/3 = 882.9597 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~5/3 = 882.8731 ¢

Optimal ET sequence: 3, 4, 11, 15, 19, 34, 53, 87, 140

RMS error: 0.2444 cents

2.9/5.… subgroups

Kiribati

See also: Chromatic pairs § Kiribati

Kiribati is related to nakika as well as to octacot.

Subgroup: 2.9/5.7/3.11/9

Comma list: 100/99 ([2 -2 0 -1⟩), 245/242 ([-1 -1 2 -2⟩)

Subgroup-val mapping: [⟨1 1 1 0], ⟨0 -2 3 4]]

Gencom mapping: [⟨1 1/10 -4/5 11/10 1/5], ⟨0 -3/2 -1 3/2 1]]

mapping generators: ~2, ~21/20

Optimal tunings:

  • Subgroup WE: ~2 = 1199.6137 ¢, ~21/20 = 87.8636 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~21/20 = 87.8385 ¢

Optimal ET sequence: 13, 14, 27, 41

RMS error: 1.245 cents

Terrain

"Terrain" redirects here. For the scale, see Terrain (scale).
See also: Chromatic pairs § Terrain

Terrain, the 6 & 21 temperament in the 2.9/5.9/7 subgroup, tempers out the landscape comma and is related to domain. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

Subgroup: 2.9/5.9/7

Comma list: 250047/250000 ([-4 6 -3⟩)

Subgroup-val mapping: [⟨3 0 -4], ⟨0 1 2]]

mapping generators: ~63/50, ~9/5

Optimal tunings:

  • Subgroup WE: ~63/50 = 400.0012 ¢, ~9/5 = 1017.5415 ¢ (~10/9 = 182.4620 ¢)
  • Subgroup CWE: ~63/50 = 400.0000 ¢, ~9/5 = 1017.5388 ¢ (~10/9 = 182.4612 ¢)

Optimal ET sequence: 6, 21, 27, 33, 105, 138, 171, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558, 3729, 3900, 7971

RMS error: 0.00844 cents

2.75.… subgroups

Archagall

See also: Fifthplus

By tempering out the comma 24576/24565 in the 2.75.85 subgroup, we have three 85/64's up and one octave down as a 75/64 and we have two 128/85's up and one octave down as a 17/15 whole tone. It is because of this combination of accuracy, efficiency and mapping-wise simplicity and its corresponding explanatory power in what this comma does that the comma has been named the archagallisma. The MVP stands for minimum viable product, as this is the core of what the archagall logic achieves, with further extensions adding to the subgroup while avoiding significantly impacting its accuracy. This is a highly accurate temperament that could be considered to be encoding the "high-accuracy logic" of superpyth and which is inescapably related to the 17L 5s scale form as it is the 17 & 22 temperament (or less accurately, the 5 & 17 temperament) in the 2.75.85 subgroup.

It is perhaps worth noting that 83edo is approximately an optimal tuning for getting all of 75/64, 85/64, and the interval between them, 17/15, close to just, in that larger tunings compromise on tone-efficiency.

Subgroup: 2.75.85

Comma list: 24576/24565 ([13 1 -3⟩)

Subgroup-val mapping: [⟨1 2 5], ⟨0 3 1]]

mapping generators: ~2, ~85/32

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9692 ¢, ~85/64 = 491.5853 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~85/64 = 491.5794 ¢

Optimal ET sequence: 5, 12, 17, 22, 61, 83, 310, 393, 476, 1345, 1821, 4118…, 5939…, 7760…

2.75.9/7.85 subgroup

A fairly natural way to extend archagall is by tempering out 2025/2023 (S15/S17), which equates a stack of two 17/15's with 9/7 without much damage. As 9/7 was not previously in the subgroup, this does not decrease the rank of the temperament and qualifies a proper and natural extension. We can equally get the same temperament by tempering out S15/S16 instead (equating a stack of three 16/15's with 17/14); however, 16/15 is not in the subgroup, so it is preferred to think of it as adding 2025/2023.

Subgroup: 2.75.9/7.85

Comma list: 2025/2023 ([2 -2 1 0⟩), 24576/24565 ([13 1 0 -3⟩)

Subgroup-val mapping: [⟨1 2 6 5], ⟨0 3 -4 1]]

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0241 ¢, ~85/64 = 491.3358 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~85/64 = 491.3290 ¢

Optimal ET sequence: 5, 12, 17, 22, 83, 105, 127, 403, 530, 657, 784, 1441…

2.7/3.… subgroups

Mothwelltri

See also: Chromatic pairs § Mothwelltri

Mothwelltri, the 4 & 5 temperament in the 2.7/3.11 subgroup, tempers out 99/98 and is related to orwell. The tonic and the first two generator steps make a mothwellsmic triad, hence the name.

Subgroup: 2.7/3.11

Comma list: 99/98 ([-1 -2 1⟩)

Subgroup-val mapping: [⟨1 0 1], ⟨0 1 2]]

Gencom mapping: [⟨1 0 0 0 1], ⟨0 -1/2 0 1/2 2]]

mapping generators: ~2, ~7/3

Optimal tunings:

  • Subgroup WE: ~2 = 1200.3175 ¢, ~7/6 = 273.2462 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~7/6 = 273.4280 ¢

Optimal ET sequence: 4, 9, 13, 22, 79

RMS error: 1.064 cents

2.7/5.… subgroups

Tridec

See also: Chromatic pairs § Tridec
See also: Non-over-1 temperament § Tridec

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends #Petrtri.

Subgroup: 2.7/5.11/5.13/5

Comma list: 847/845 ([0 1 2 -2⟩), 1001/1000 ([-3 1 1 1⟩)

Subgroup-val mapping: [⟨1 -2 3 2], ⟨0 4 -3 -1]]

Gencom mapping: [⟨1 0 -3/4 -11/4 9/4 5/4], ⟨0 0 0 4 -3 -1]]

mapping generators: ~2, ~20/13

Optimal tunings:

  • subgroup WE: ~2 = 1200.0343 ¢, ~20/13 = 745.4658 ¢
  • subgroup CWE: ~2 = 1200.0000 ¢, ~20/13 = 745.4431 ¢

Optimal ET sequence: 5, 8, 21, 29, 37, 66

RMS error: 0.1613 cents

Naiadec

Subgroup: 2.7/5.11/5.13/5.17/5

Comma list: 170/169 ([1 0 0 -2 1⟩), 221/220 ([-2 0 -1 1 1⟩), 847/845 ([0 1 2 -2 0⟩)

Subgroup-val mapping: [⟨1 -2 3 2 3], ⟨0 4 -3 -1 -2]]

Gencom mapping: [⟨1 0 -3/4 -11/4 9/4 5/4 9/4], ⟨0 0 0 4 -3 -1 -2]]

mapping generators: ~2, ~20/13

Optimal tunings:

  • Subgroup WE: ~2 = 1200.4068 ¢, ~20/13 = 745.3706 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~20/13 = 745.0940 ¢

Optimal ET sequence: 5, 8, 21, 29, 95*, 124*

* Wart for 17/5

RMS error: 0.7521 cents

2.25/7.… subgroups

Supramine

Supramine extends supramin and approximates the 14:17:19:23:25 pentad in just six generator steps, at the cost of some accuracy. 25edo remains a strong tuning.

Subgroup: 2.25/7.17/7.19/7.23/7

Comma list: 323/322 ([-1 0 1 1 -1⟩), 392/391 ([3 0 -1 0 -1⟩), 476/475 ([2 -1 1 -1 0⟩)

Subgroup-val mapping: [⟨1 -2 0 4 3], ⟨0 3 1 -2 -1]]

mapping generators: ~2, ~17/14

Optimal tunings:

  • Subgroup WE: ~2 = 1199.7558 ¢, ~17/14 = 335.4278 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~17/14 = 335.4885 ¢

Optimal ET sequence: 7, 18, 25, 68, 93

Badness (Sintel): 0.053

2.11/5.… subgroups

Petrtri

See also: Chromatic pairs § Petrtri
See also: 5L 3s/Temperaments § Petrtri

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup and tempers out 2200/2197. It restricts tridec. The tonic and the first two generator steps make a petrmic triad, hence the name.

Subgroup: 2.11/5.13/5

Comma list: 2200/2197 ([3 1 -3⟩)

Subgroup-val mapping: [⟨1 0 1], ⟨0 3 1]]

Gencom mapping: [⟨1 0 -1/3 0 -1/3 2/3], ⟨0 0 -4/3 0 5/3 -1/3]]

mapping generators: ~2, ~13/10

Optimal tunings:

  • Subgroup WE: ~2 = 1199.9572 ¢, ~13/10 = 454.9958 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~13/10 = 455.0039 ¢

Optimal ET sequence: 5, 8, 21, 29, 153, 182, 211, 240, 269, 509, 778…, 1287…

RMS error: 0.0749 cents

Other 2-repeating subgroups

Supramin

Supramin restricts supramine and is a remarkable low-complexity microtemperament that contains the 14:17:19 triad within just four generator steps. An excellent tuning is 25edo, which provides an accurate yet tone-efficient tuning of this temperament. It was named by Overthink in 2026 after the fact that the generator is a 17/14 supraminor third, two of which reach 28/19. It is related to cohemimabila.

Subgroup: 2.17/7.19/7

Comma list: 5491/5488 ([-4 2 1⟩)

Subgroup-val mapping: [⟨1 0 4], ⟨0 1 -2]]

mapping generators: ~2, ~17/7

Optimal tunings:

  • Subgroup WE: ~2 = 1200.0220 ¢, ~17/14 = 335.7933 ¢
  • Subgroup CWE: ~2 = 1200.0000 ¢, ~17/14 = 335.7852 ¢

Optimal ET sequence: 7, 18, 25, 118, 143, 168, 193, 218, 461, 679, 1576, 2255…

Badness (Sintel): 0.005

3/2.5/2.… subgroups

Main article: Half-prime subgroup

Hemihemi

Named by CompactStar in 2023, hemihemi tempers out 10976/10935, the hemimage comma, in the 3/2.5/2.7/2 subgroup.

Subgroup: 3/2.5/2.7/2

Comma list: 10976/10935 ([-7 -1 3⟩)

Subgroup-val mapping: [⟨1 2 3], ⟨0 3 1]]

mapping generators: ~3/2, ~28/27

Optimal tunings:

  • Subgroup WE: ~3/2 = 702.6069 ¢, ~28/27 = 60.4665 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~28/27 = 60.8011 ¢

Optimal ET sequence: *11, *12, *23, *35, *58, *93, *151†, *244†

* Wart for 3/2
† Wart for 7/2

Halftone

Main article: Halftone

Named by CompactStar in 2023, halftone tempers out 9604/9375 in the 3/2.5/2.7/2 subgroup.

Subgroup: 3/2.5/2.7/2

Comma list: 9604/9375 ([-1 -5 4⟩)

Subgroup-val mapping: [⟨1 3 4], ⟨0 -4 -5]]

mapping generators: ~3/2, ~15/14

Optimal tunings:

  • Subgroup WE: ~3/2 = 702.6568 ¢, ~15/14 = 128.2849 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~15/14 = 128.8183 ¢

Optimal ET sequence: *5, *11, *27, *38

* Wart for 3/2

3/2.5/2.7/2.11/2 subgroup

Subgroup: 3/2.5/2.7/2.11/2

Comma list: 1232/1215 ([-5 -1 1 1⟩), 9604/9375 ([-1 -5 4 0⟩)

Subgroup-val mapping: [⟨1 3 4 4], ⟨0 -4 -5 1]]

Optimal tunings:

  • WE: ~3/2 = 704.8651 ¢, ~15/14 = 130.9114 ¢
  • CWE: ~3/2 = 701.9550 ¢, ~15/14 = 129.6376 ¢

Optimal ET sequence: *5, *11, *16, *27…, *43…, *70…

* Wart for 3/2

3/2.5/2.7/2.11/2.13/2 subgroup

Subgroup: 3/2.5/2.7/2.11/2.13/2

Comma list: 275/273 ([-1 2 -1 1 -1⟩), 1232/1215 ([-5 -1 1 1 0⟩), 1323/1300 ([3 -2 2 0 -1⟩)

Subgroup-val mapping: [⟨1 3 4 4 5], ⟨0 -4 -5 1 -2]]

Optimal tunings:

  • WE: ~3/2 = 702.9683 ¢, ~15/14 = 130.0424 ¢
  • CWE: ~3/2 = 701.9550 ¢, ~15/14 = 129.6034 ¢

Optimal ET sequence: *5, *11, *16, *27†

* Wart for 3/2
† Wart for 11/2

Semiwolf

Semiwolf provides an interpretation for the 3L 2s⟨3/2⟩ mos scale. The name semiwolf comes from two ~7/6 generators approximating a 27/20 wolf fourth. This is further extended to the 11-limit in two interpretations: semilupine, where 2 major 2-mossteps (LL) equal 11/9, and hemilycan. where 1 major and 2 minor 2-mossteps (sLs) equal 11/9. Basic 8edf fits both extensions.

Subgroup: 3/2.5/2.7/4

Comma list: 245/243 ([-5 1 2⟩)

Subgroup-val mapping: [⟨1 1 2], ⟨0 2 -1]]

mapping generators: ~3/2, ~9/7

Optimal tunings:

  • Subgroup WE: ~3/2 = 703.7414 ¢, ~9/7 = 440.9014 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~9/7 = 440.8307 ¢

Optimal ET sequence: *8, *43, *51, *59, *67, *75…, *83…

* Wart for 3/2

Hemilycan

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 245/243 ([-5 1 2 0⟩), 441/440 ([2 -1 2 -1⟩)

Subgroup-val mapping: [⟨1 1 2 5], ⟨0 2 -1 -4]]

Optimal tunings:

  • Subgroup WE: ~3/2 = 703.2870 ¢, ~9/7 = 441.1967 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~9/7 = 440.4501 ¢

Optimal ET sequence: *8, *28, *35, *43, *51, 110…

* Wart for 3/2

Semilupine

Subgroup: 3/2.5/2.7/4.11/4

Comma list: 100/99 ([-3 2 0 -1⟩), 245/243 ([-5 1 2 0⟩)

Subgroup-val mapping: [⟨1 1 2 0], ⟨0 2 -1 4]]

Optimal tunings:

  • Subgroup WE: ~3/2 = 704.6266 ¢, ~9/7 = 439.2433 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~9/7 = 438.6989 ¢

Optimal ET sequence: *5, *8, *45…, *53…, *61…, *69…, *77…

* Wart for 3/2

3/2.5/4.… subgroups

Ambulatory

Ambulatory tempers out 686/675 in the 3/2.5/4.7/4 subgroup and is generated by a sharp ~15/14. This implies a stack of two generators represents 7/6, and a stack of three generators represents 5/4.

Ambulatory can be considered the 3/2.5/4.7/4 analog of BPS or meantone, using 5:6:7 as its fundamental consonant chord in the place of 3:5:7 or of 4:5:6. Ambulatory is actually similar to meantone in several ways:

  • They are both of low badness.[dubious – discuss]
  • In most optimal tunings, meantone tunes 3/2 flat and 5/4 slightly sharp, while ambulatory tunes 7/5 flat and 6/5 slightly sharp.
  • Meantone[7] has 5/4 and 6/5 share a scale degree. Ambulatory[5] has 6/5 and 7/6 share a scale degree.
  • Meantone[7] is a tempering of the 7-note AGS(5/4, 6/5) (Ptolemy's intense diatonic) with the large and medium steps equated. Ambulatory[5] is a tempering of the 5-note AGS(6/5, 7/6) (assuming an equivalence of 3/2) with the medium and small steps equated.

Ambulatory was first documented by CompactStar who originally proposed the name emergency, which refers to the North American emergency telephone number 911, because 9\11edf is a possible generator. This was deemed as too dark so the name ambulatory" was suggested by Frostburn to more subtly reference the same thing.

Subgroup: 3/2.5/4.7/4

Comma list: 686/675 ([-3 -2 3⟩)

Subgroup-val mapping: [⟨1 0 1], ⟨0 3 2]]]

mapping generators: ~3/2, ~15/14

Optimal tunings:

  • Subgroup WE: ~3/2 = 701.8452 ¢, ~15/14 = 129.7976 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~15/14 = 129.5324 ¢

Optimal ET sequence: *5, *11, *27, *38

* Wart for 3/2

Poseidon

Todo: rename, resolve temperament name conflict

Subgroup: 3/2.5/4.11/8

Comma list: 121/120 ([-1 -1 2⟩)

Subgroup-val mapping: [⟨1 1 1], ⟨0 -2 -1]]]

mapping generators: ~3/2, ~12/11

Optimal tunings:

  • Subgroup WE: ~3/2 = 701.8452 ¢, ~12/11 = 157.8337 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~12/11 = 157.8574 ¢

Optimal ET sequence: *4, *5, *9, *22, *31, *40

* Wart for 3/2

Other 3/2-repeating subgroups

Auk

Auk tempers out the comma 87808/85293 in the 3/2.7.13 subgroup, equating three instances of 7/1 (fifth-reduced to 112/81) with 13/1 (fifth-reduced to 832/729). The generator when reduced is a superfourth in the neighborhood of 550 cents, representing 112/81. It possesses nontrivial mos of the families 1L 3s⟨3/2⟩ ("neptunian"), 4L 1s⟨3/2⟩, 5L 4s⟨3/2⟩, and 9L 5s⟨3/2⟩. 9edf is a particularly good equal temperament that supports it. It contains tempered versions of the chords 1–832/729–112/81 (fifth-reduction of 1:7:13) and 1–63/52–112/81 (fifth-reduction of 1/(13:7:1)).

Subgroup: 3/2.7.13

Comma list: 87808/85293

Subgroup-val mapping: [⟨1 0 -8], ⟨0 1 3]]

mapping generators: ~3/2, ~7

Optimal tunings:

  • Subgroup WE: ~3/2 = 706.5197 ¢, ~112/81 = 538.8834 ¢
  • Subgroup CWE: ~3/2 = 701.9550 ¢, ~112/81 = 543.8799 ¢

Optimal ET sequence: *4, *13, *17df, *21df, *38ddff, *59ddffff

* Wart for 3/2

Doubleton

Subgroup: 3/2.7.13

Comma list: 1352/1323 ([-3 -2 2⟩)

Subgroup-val mapping: [⟨2 0 3], ⟨0 1 1]]

mapping generators: ~26/21, ~7

Optimal tunings:

  • Subgroup WE: ~26/21 = 352.6518 ¢, ~112/81 = 552.6017 ¢
  • Subgroup CWE: ~26/21 = 350.9775 ¢, ~112/81 = 565.7161 ¢

Optimal ET sequence: *4, *10, *14f, *46dfff

* Wart for 3/2

5/2-repeating subgroups

Hyperion

Subgroup: 5/2.7.11

Comma list: 341796875/329832448 ([11 1 -5⟩)

Subgroup-val mapping: [⟨1 -1 2], ⟨0 5 1]]

mapping generators: ~5/2, ~44/25

Optimal tunings:

  • Subgroup WE: ~5/2 = 1582.2958 ¢, ~44/25 = 990.1327 ¢
  • Subgroup CWE: ~5/2 = 1586.3137 ¢, ~44/25 = 991.6570 ¢

Optimal ET sequence: *8, *83…, *91…, …, *155…, *163…

* Wart for 5/2

See also

References

  1. ↑ Yahoo! Tuning Group | Ridiculously awesome 6&11 2.3.7.11.13 subgroup temperament
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